What is the difference between kiedy and gdy?
There are pretty close in meaning. The most important difference is that you can only use "kiedy" in questions about time.
Why is the to necessary?
Maybe it's not absolutely necessary, but I think it makes the sentence more natural. I'd say that it shows that the second part of the sentence is a result of the first part.
I'm wondering why "Jak śpię to nie movię" wouldn't be correct.
It should be „mówię” at the end. I don't know if the course accepts that, but I often speak like that.
I think it's a bit colloquial, but seems fine to me (apart from what Vengir pointed out). Added now.
Why not also: Nie mowie kiedy spie?
It works already.
"Nie mówię kiedy śpię” - Well, nothing wrong in here but it's not the preferred way of expressing this in Polish language.
Nie mówię kiedy śpię
Nie mówię gdy śpię
"Nie mowie podczas gdy spie." was listed as correct. Does podczas always have to be used with gdy?
With normal verbs (clauses), yes. But you don't add „gdy” if you use „podczas” with nouns or gerund verbs.
I guess "nie mówię przez sen" isn't a literal translation, but is that not a set phrase in Polish? Could it work here?
I believe that's more like "I do not talk in my sleep", but sure, why not. Added.
Would it be wrong to have "Nie mówię kiedy śpię."
That's a perfectly fine sentence. Well, it needs a comma before "kiedy", but that's the only problem. It works.