"Infinite regress of causes and effects can't be real/ is illogical/impossible".
- Why assume it is impossible. (Reality doesn't have to appeal to our common sense)
- Why assume we understand what it actually IS as finite creatures.
- Why assume that it is the only alternative to "uncaused cause"? Maybe there are million other alternatives we can't even begin to fathom as they don't touch "our reality".
I will make some preliminary remarks, and then answer your three individual questions. As Conifold points out in the comments, this issue can only really be understood by understanding the Münchhausen trilemma, which is itself a logical consequence of the law of the excluded middle. Suppose you have a set of things, and a binary relation < (e.g., representing causality) where each thing can either be related to one other thing by that binary relation, or not related to any other thing by that binary relation. So, for example, thing A either has A < B for some B, or it does not have A < B for any B. Now suppose that you try to trace this relation from some starting item until it terminates. The Münchhausen trilemma tells us that there are three possibilities: (1) a circularity, where you get back to a previous thing: A < B < C < D < A < ... (2) an infinite regress, where you get a different thing each time and the series never terminates: A < B < C < D < ... ; or (3) the series terminates: A < B < C < D.
To apply the Münchhausen trilemma in this context, start at some caused thing. Either this had a cause or it did not. If it had a cause, go to its cause. Either that had a cause or it did not. If it had a cause, go to its cause. Either that had a cause or it did not, and so on. This inquiry must either terminate by finding a thing that had no cause, or it does not terminate. If it does not terminate, this either occurs because you get back to a previous cause and then cycle through a finite number of causes circularly, or you don't get back to a previous cause, and so you have an infinite chain of different causes. Now, if we exclude the circular case by viewing this as not really being causality at all (because each thing ends up being asserted to be the cause of itself), we are left with the other two alternatives: either there is something at the base that had no cause, or there is an infinite regress of cause and effect.
The Münchhausen trilemma basically leads different schools of philosophy to branch off into different understandings of metaphysics and epistemology, depending on which of these things they think makes sense in the context of metaphysical and epistemological questions. Some schools of philosophy have no issues with infinite regresses of causes and effects, and others regard that as an indication of incorrect reasoning.
In any case, regardless of which particular school of philosophy you look at, if you accept the law of the excluded middle, and thus accept the Münchhausen trilemma, you are going to have to grapple with this trichotomy. There could certainly be millions of things that we do not fathom, but each of those things must logically either be causes of a particular thing or not. If you begin from the premise that you cannot reason about the available logical alternatives in a situation ---even to the extent of just applying the law of the excluded middle leading to the Münchhausen trilemma --- because there may be things that "we can't even begin to fathom", then your whole epistemology is a non-starter. This seems to be what you are saying when you assert that our conceptual framework breaks down and that certain things are simply unspeakable. That would not just impede reasoning about theism; it would seem to me to be a form of nihilism, which essentially means that humans can never reason validly, and can never know anything about anything.
- Why assume it is impossible. (Reality doesn't have to appeal to our common sense)
I don't think people necessarily assume this is impossible. There are some schools of philosophy that accept "infinitism" in the metaphysical realm, and would not be disturbed by an assertion of an infinite regress of causes and effects. Those schools tend to adopt this approach due to an aversion to the other two cases; they view a terminating uncaused thing as wrong, and they view a circularity of causes as wrong.
- Why assume we understand what it actually IS as finite creatures.
From predicate logic and the foundations of mathematics, we have a pretty good understanding of infinity, so I don't agree that we are "finite creatures" in the sense of being unable to properly understand the concept of infinity. To say that there is an infinite regress of causes and effects is equivalent to saying that the regress of causes and effects does not terminate in any finite number of steps, and there is no circularity of causes. So, if we can understand the finite whole numbers (i.e., the natural numbers), and we understand logical negation and the universal operator, then we understand infinity in this context.
- Why assume that it is the only alternative to "uncaused cause"? Maybe there are million other alternatives we can't even begin to fathom as they don't touch "our reality".
As shown above, this is not an assumption at all. If one accepts the law of the excluded middle, this leads to the Münchhausen trilemma. If one then rules out circularity of cause as not really being a valid form of causation at all, then one is left with the two other options, and thus has the dichotomy of an infinite regress of causes or an uncaused thing at the base. (Note from the Objectivist school that the uncaused thing at the base might not be a cause. The Objectivists argue that only actions pertaining to things have causes, and so the thing at the base is an uncaused thing, and its actions are caused by its nature, which is uncaused. This opens a can of worms that gets off-topic here, so I won't elaborate beyond that.)