One of the weaknesses of the theory of evolution is that it has no coherent explanation of what "chance" is. Indeed, "chance" is difficult to rigorously define. If it is not defined, how is anyone to know whether it exists? Has it ever been empirically observed?
One place where we can observe what most people would agree is chance is . . . at the casino. And there we see that to generate "chance" or "randomness" requires a careful construction of various apparatuses, the common purpose of which is to insulate the outcome from the will of an individual. Whether it is a roulette wheel, a pair of dice, or a deck of cards, the essential function of a game of chance is to prevent the operator from influencing the outcome.
Now then, if "chance" is indeed a construct to insulate phenomena from will, and if we were ever to observe "chance" occurring in nature, we might have to conclude that God exists. The evidence, i.e., the existence of "chance," would imply that there is a will from which the phenomena of nature are being insulated.
There is not space here, nor do I have the time now to go into this more deeply. But if you will try as an exercise to rigorously define "chance" in such a way as to take that definition and use it as a guide to empirical observation, as any good scientist should do, you may find that the above reasoning is not easily dismissed.
In my view it ironic that the postulate of "chance" in the theory of evolution is an implicit acknowledgement of God's existence. But others have made the above point in other language.
Sure. Take a particle with a known half-life. In a time interval equal to the half-life, there is a 50% chance that it will decay. There is no way to predict the outcome in advance.
Your reasoning, as I understand it, is that randomness is "unnatural" and thus its existence implies a creator. You support the premise by noting that human beings often go to a great deal of effort to produce randomness in certain events. While this is true, I don't believe it follows from that that chance would not be found in nature in the absence of a creator. There are plenty of things that occur in nature that are difficult for humans to replicate, nuclear fusion for example.
This is rather tangential to the evolution question. As it stands now, evolution is the theory that best fits the facts. Evolution is not incompatible with the existence of God, only with certain literal interpretations of the Bible.