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Let's look at the first group of "refutations" that you posted My comments in boldface:

[Argument Against] The Argument from Motion

St Thomas subscribed to Aristotle's notion of a Prime Mover. For Aristotle this Prime Mover was the fifth in a set of elements. It was motionless but caused the movement of the previous four, these being: earth, fire, air and water. Straight away, you see where St Thomas is coming from.

Motion -- whether it be physical movement, or a change in temperature -- cannot have started on its own because nothing can be reduced from potentiality to actuality, unless by something else which is in a state of actuality. Wood, for example, has the potential to be hot but cannot itself become hot without fire which is actually hot.

Now, in the same way that fire cannot be actually and potentially hot at the same time (because when it's hot it's potentially cold) something cannot be both mover and moved: a thing cannot move of itself. If there is motion in a thing, some thing must have caused it to move. And if that thing was also in motion when it caused motion in the first thing, then something must have caused its motion. Obviously, you can't go on forever, so the argument is made that there must have been a first mover, itself motionless and causing all other forms of motion.

In thinking circles, this is called Affirming the Consequent. St Thomas has not yet demonstrated god exists but he is "proving" God's existence by describing God's qualities. Why? (Note here the author does not reject the idea of a Prime Mover, but rather, the ascribing of the quality of Prime Mover to God) Well, in opposition to Aristotle, St Thomas saw all motion as purposeful. A purpose denotes a plan and, if there's a plan, there must be planner. Therefore movement in the universe happening to the scale at which it does "proves" a very big planner indeed.

(No. St. Thomas invokes a Planner or Designer in his argument from Design. In his argument from motion, he derives the idea of a Prime Mover).

Aristotle didn't see movement as having an end as designed by god. His god did not appoint ends: they just were. Motion, to Aristotle, and to many atheists, can just be motion, with no particular plan behind it.

Again, St. Thomas does not infer purpose in his argument from motion. He proves that a Prime Mover is logically necessary. The alternative is that motion stretches back infinitely in sequence. However, although an infinite sequence can exist theoretically, an infinite sequence cannot exist actually because in becoming actual the series becomes finite. Ultimately, something must have been set in motion by a Prime Mover.

Now, is the Prime Mover God? We know from St. Thomas' fifth argument from degrees of perfection that "the most perfect being exists." A perfect being lacks nothing. So, logically, if there is a Prime Mover, the Prime Mover must be The Most Perfect Being, who everyone calls God.

Now, the author also misrepresents St. Thomas' idea of motion, which is much better presented here:

The Problem of Change
The basic notions of Aristotle's philosophy of nature can be understood from his analysis of change. When Aristotle undertook to explain how it is that things change, a fact apparent to anyone, he had first to confront the seemingly iron-clad logic of Parmenides. Bound by this logic, Parmenides had been forced to the position that there is in reality no change at all. All change is mere appearance; reality is One, and this One, which only is, is unchanging. He was forced to this position because, as he understood the terms of the problem, change is logically not possible. Not having the notion of potency, Parmenides had argued that there are only two alternatives for anything, being and non-being. No new being can come from non-being since "nothing comes from nothing." Nor can new being come from being since what has being, already is and does not begin to be: "being cannot come from being since it is already."
The advance that Aristotle made over Parmenides consists in seeing that, although it is true that "nothing can come from nothing," it is not entirely true that "being cannot come from being." One must distinguish being-in-act from being-in-potency. While it is true that from being-in-act, being-in-act cannot come since it would already be. The alternative from which being can come is not non-being, but being-in-potency. From being-in- potency there can come being-in-act.
Accidental Change
Michelangelo is alleged to have said that when he set out to sculpt a statue from a formless block of marble, he sought only to remove the excess marble from the statue that was already there inside the block. This sentiment expresses what Aristotle discovered to be necessarily true for all change.
Aristotle discovered the concept of potency by observing accidental changes. He observed, for instance, that a sculptor can make a statue from a block of marble. This is possible only because the block of marble is endowed with a certain property - the possibility and capacity of being transformed. The figure of the statue is in potency in the block of marble. This potency is not nothing, it is not non-being. It is real; not with the reality of being-in-act, but with the reality which corresponds to being-in- potency.
The first principles of motion can be discovered but they cannot be demonstrated. In order to demonstrate them, we would have to assume that they are the result of other principles, in which case they would not be the first principles. (Posterior Analytics I, 3) These principles are not demonstrated but discovered by analyzing substantial changes.

Similarly, who is to say there was only one original Prime Mover? Why not two? Because two things, different in substance, cannot be the First Mover. The Prime Mover must be one substance. Interestingly, that does not rule out the possibility of a multi-personal substance like the TrinityWhy not a whole team of gods, working on the project together? See previous commentPerhaps our universe is one of many attempts, some good, some botched.No. It is logically impossible for The Most Perfect Being to errOr our universe could be, in David Hume's words, the poor first attempt "of an infant deity who afterwards abandoned it, ashamed of his lame performance."See previous comment

Strangely, St Thomas thought that this was the most self-evident, self-proving of his "proofs". Partly this is because humans are rational beings. They have reason-seeking minds and they habitually assume that everything has a reason. It is hard for humans to accept that some things may not have a reason, but there might be instances where this is actually the case. This assertion itself is attempt to use reason(!) although it is irrational.St Thomas did not account for this possibility because to do so would have meant challenging fundamental elements of his worldview. Unlike atheists, Thomas certainly was willing to acknowledge the "super-rational." However, anything that exceeds Reason, such as the divinely revealed Trinity, cannot contradict Reason, since The Most Perfect Being is the author or Faith and Reason.

* The Argument from the Nature of the Efficient Cause

Proof 2 is a furtherance of the notion that the first cause was also, in philosophical language, "efficient". That is, the Prime Mover has the power to make a change. This is an extremely debated and deep philosophical question. So? It ties into notions of cause and effect and the human ability to separate the two as necessarily having any connection.

Basically, the second proof is that there is no case known where a thing is found to be the efficient cause of itself: because then it would have to come before itself which is impossible because cause and effect have a particular order. Sounds crazy, doesn't it? That a thing cannot bring itself into existence? That a thing can't exist before it exists? A match bursts into flames after it has been struck, not before. To take away the cause is to take away the effect. No striking, no flames. If something didn't start the universe then the universe wouldn't exist, so something must have started the universe. There must have been a first efficient cause.

It all makes sense up to this point but then St Thomas decides, arbitrarily, that this first cause is called "God" by everyone. He's sort of saying, "Oh, you know what I mean." Because this proof is based heavily on acceptance of the first proof, he is again assuming that motion has purpose, and that purpose denotes an intelligent planner. You have to decide here, as you did in the first proof, if you believe that motion always has a purpose.

The Most Perfect Being lacks no power or perfection. Therefore The Most Perfect Being must be The First Cause, which everyone calls God.

* The Argument from Possibility and Necessity

Things are generated, they exist for a certain period of time, and then they perish. If everything was like this then one day nothing would exist. However, stuff patently does exist. And if stuff can only exist through being the product of stuff that already exists, the argument is made that because stuff exists now, stuff must always have existed.

Once again, this regress obviously cannot go on forever and so it's proposed that there are two types of beings: those that owe their existence to what came before them, and those that don't. There must be at least one being who did not owe its existence to something that came before it. Beings are normally necessary to make other beings but such a being has its own necessity. This very first being, this first piece of stuff, is called God.

This argument is based on another extremely wobbly assumption tackled on the very top of the Basic Arguments page, under the heading No Need for a First Cause. Just to remind you, one can choose to believe in a super-human, all-knowing, all-powerful god who has existed for eternity and who made the universe and everything in it, or one can choose to believe that the universe just popped into existence for no intelligent reason in particular.

Yes, one can choose to believe irrational ideas. But irrational ideas are false.

If you ask the question "Who made the universe?" it is also reasonable to ask the question "Who made god?"

No, it is not. This is a common logical "category error." God is not a thing in the sense of most things that we experience. Most things are substances that are ultimately composed of two things: existence and act of existence. For example, I can imagine a unicorn, but a unicorn does not have actual existence. Or, I can imagine my deceased dog, but my deceased dog does not have actual existence. Everything we see around us exists and is in the act of existence. In other words, what a thing is is different from whether or not it exists. But God is fundamentally different. For God, essence and act of existence are the same. God is Being itself. He is the one, utterly simple (He has no parts), perfect spiritual being. We can know this with certainty from the argument from degrees of perfection. (One part would be superior to the other. Similarly, composed things can be decomposed, which is an imperfection)

Basing your belief system on the answer to the former of these questions is irrational if you cannot also answer the latter.

And, of course, if you can come to the conclusion that at least one being came into existence of its own necessity, why not two, three or a larger team?

See comment above. There cannot be more than one most perfect being

And then there's the question of infinity. If there are only contingent beings (those beings that are capable of ceasing to exist) and this universe has existed through an infinite amount of time, then all possibilities of everything must have already occurred because there's been an infinite amount of time for all possibilities to happen. And one of those possibilities is the simultaneous non-existence of all beings. So, how come you're still here reading this?

That is the argument

Mind you, this only rings true if all possibilities must occur within a certain period of time which is patently untrue.

Hello. Critics of the First Cause argument assume that the universe has existed for an infinite amount of time. By definition, all possibilities must have been realized. That makes the rest of this paragraph utter nonsense. It must happen some time that all contingent beings do not exist, but it cannot have happened yet because, patently, we do exist and we can't have come from nowhere twice! But then, if it has yet to happen, that means we must have always been around, which is also absurd. You can see that the third proof is so fraught with paradoxes, contradictions, absurdities and conundrums that it really doesn't prove anything. It's a good one for parties though.

No, the author is fraught with paradoxes, contradictions, absurdities and conundrums that he really doesn't prove anything.

* The Argument from Gradation [Degrees of Perfection]

If we take two things and say that X is better than Y then X becomes the "best". X is now held to cause all imperfections in whatever is less good than X.

(Straw man. This is not the argument. The argument is that if good and better exists, for example, there must exist an ultimate "best." If there is no "best," then logically there can be no continuum)

X might be truth, nobility, goodness, etc. Reversing this logic leads us to conclude that "there is some cause of existence and goodness and whatever other perfections are characteristic of things, and this we call God."

Again, a straw man. If good and better beings exist, then there must exist a Most Perfect Being, who everyone calls God.

But that's just plain nutty. The next-to-best tennis ball in the universe is not made by the best tennis ball in the world. (Complete mischaracterization of the argument) It's made by a machines in a tennis ball factory. Frequently, children turn out to be far better in many respects than their parents.(Straw man or red herring, take your pick.)

St Thomas here is caught in a web of semantics (as is much of philosophy). He has confused human descriptions of perceptions of things with the objective actuality of things. Again, close but no cigar.

The author wouldn't know an argument if it jumped up and bit him on his butt.

* The Argument from Governance [Argument from Design]

Based on notions "proved" in Proofs 2 and 3, the fifth proof proffers the notion that there must be something in the universe which is the source of all good, something by which all natural things are directed to their end, in much the same way as inanimate objects are directed by humans, e.g. an arrow shot to its mark by an archer.

Here, albeit in a roundabout way, St Thomas again affirms the consequent. "Good" is a notion that is dependent on the opposite notion of "bad" and when you say something is "good" or "bad" (in the sense that St Thomas meant it) you automatically infer that something also exists which can tell the difference between the two.

This "proof" is flawed because in its attempt to prove X it includes a quality of X as part of its proof. It does not say, for example, if A and B are both true it is rational to infer that C is also true. A more clumsy argument of this sort is the old "the bible is true because it say so" kernel.

Many atheists and agnostics do not subscribe to normal religious definitions of "good" and "bad". Actions, rather than being right or wrong, have consequences for which one must accept responsibility.

Where to begin? St. Thomas' argument is simple. When we see something that is designed, we infer a designer. For example, when I walk down the street and find a candy wrapper, I don't think, "Gee, look at the collection of processed wood and ink which happened to come together in this interesting pattern." No, I think, "I wonder where this candy wrapper came from." Similarly, we observe the harmoniously functioning, infinitely complex world around us, and infer from this incredible design and ultimate Designer.

If the author could argue about designers existing between the design and the Ultimate Designer (angels perhaps?), but there must exist an Ultimate Designer, since the Most Perfect Being must be the Ultimate Designer.

The absurd, alternative to believing in an Ultimate Designer is to believe that Design came into existence without a Designer.