But his premise says the all existence is caused; the uncaused, therefore, cannot exist.
Whoa! Where does "outside time" come from? Aquinas doesn't mention time.
The problem is not in the logical conclusion, but in the fact that the logical conclusion refutes itself.
This would of course mean that nothing exists, which is refuted because we exist.
In other words, something is wrong in his formulation. Either the premise, or the conclusion.
It would help to see what specifically your referring to in Summa.
Aquinas is using Aristotle terms for "causes" so it helps to understand efficient causes for example.
Anyway, the argument is there are first, intermediate and final causes. The "sense world" is comprised of intermediate and final causes - there must be a first cause. Nothing can be the cause of itself. The first cause cannot be a dependent cause (caused by something else). Therefore there must exist a first cause that is neither caused by itself nor caused by anything outside itself; ergo uncaused.
I don't believe there is an error in the logic of his argument, but would need to see more precisely where you are seeing one.
Not in the first cause argument, which is one of five including first mover. There's a whole section in Summa on "Whether God is eternal" in the form of objections and answers. I won't paste the whole thing here, but here's a part that refers to the first mover part of the cosmological argument of which the first cause argument is also a part.
The idea of eternity follows immutability, as the idea of time follows movement, as appears from the preceding article. Hence, as God is supremely immutable, it supremely belongs to Him to be eternal.
I should have noted that “move” for Aquinas is used the same as we use “change.” So in essence - in this part of “Whether God is eternal” - he is saying there is no time if there is no change.