To posit God simply based on a lack of explanation is known as the “God of the gaps” fallacy. A proper example would indeed be believing that gods cause lightning due to a lack of other explanations. I agree this is not proper reasoning.
However, this is not at all what thinking Christians do when they hold the view that God created the universe. Why?
When we conclude that intelligence created [the universe], it’s not simply because we lack evidence of a natural explanation; it’s also because we have positive, empirically detectable evidence for an intelligent cause. A message (specified complexity) is empirically detectable. When we detect a message like “Take out the garbage, Mom” or 1,000 encyclopedias we know that it must come from an intelligent being because all of our observational experience tells us that messages come only from intelligent beings. Every time we observe a message, it comes from an intelligent being. We couple this data with the fact that we never observe natural laws creating messages, and we know an intelligent being must be the cause. That’s a valid scientific conclusion based on observation and repetition. It’s not an argument from ignorance, nor is it based on any “gap” in our knowledge.
The title of Turek’s article, and its conclusion, points out that it is skeptics who are equally guilty — perhaps unaware — of committing this fallacy. You yourself within this second question asked, “Why do you resort to ‘God did it’ for the Big Bang when a natural answer, if we ever find one, will account for it?” I’m sure you realize that this is, in essence, “We have no other explanation, therefore, we will assert that a natural eventually-found explanation must be responsible.” To say this is different from what you suggest Christians are doing is special pleading. You might argue that a natural explanation is the default because a natural explanation is always the type of explanation we (eventually) have, but that’s circular reasoning, if the thing we’re trying to prove is whether or not a super-natural (hyphen is fine) is reasonable or at least possible.
You’re probably aware of Kalam’s cosmological argument — a classic for defenders of Christian reasoning like Dr. William Lane Craig — and the ontological argument, which 11th Century’s Saint Anselm loved and which contemporary philosopher and dept chair at Calvin College Alvin Plantinga has championed in recent years. In the case of both of these arguments, it is reasoned that a universe which began to exist (and cosmologists are in complete agreement that it did) must necessarily have an ultimate cause. The idea of an infinite regress of causes is not philosophically sound, so it has to stop somewhere, and that somewhere had to have been greater and more “intelligent” than the universe it caused. What is greater than our natural universe? The super-natural.
P.S. I didn’t address this bit below, because I didn’t follow what your point was (we theists do believe what cosmologists say about the big bang and it causing the expansion of the universe), but I believe I addressed your main point in question number two
You may not know anything about neurology or meteorology, but you have no reason to believe that the specialists in these fields are lying. You accept what they say, not on faith but on trust, and anyway, an investigation into the veracity of what neurologists and meteorologists say can always be had. By contrast, you don’t trust cosmology’s current understanding of the Universe, that there was a “Big Bang” that led to the expansion of the Universe and that this happened roughly fourteen billion years ago.