1) experience in is typically followed by a field (i.e., I have experience in archaeology) while experience with is typically followed by a person, place, or thing (e.g., I have experience with children, I have experience with languages, I have experience with cars (where as you could say "I have experience in mechanics.")). I will say this though: this is excessively nit picky and I don't think anyone would bat an eye if you switched the two around (said "I have experience with archaeology" or "I have experience in cars.")
2.) The second one is grammatically correct, the first one is slightly awkward, as you're retaining the question inversion in a sentence that does not require it. I would definitely go with the second.
3.) "I'm stuck in, watching a romantic movie." This sentence actually has two clauses and is still slightly awkward, as "I'm stuck in" is implying the reason you're "watching a romantic movie." To say "I'm stuck in" in this context I'm thinking you're saying that you have to stay home for some reason (i.e., babysitting or because it's raining or your parents have grounded you) and that is why you're watching a romantic movie. If you're watching it because you're girlfriend is making you and you have no way out of the situation you might say "I'm stuck watching a romantic movie." My instinct is that this latter sentence is what you're looking for. But I'm not sure.
4.) It is 4 hours distant is correct but slightly strange-sounding. Better "It's four hours away."
5.) both compared to and compared with are correct and I can't think of any linguistic difference except to say that compared with is slightly more formal.
6.) I'd like to practice. I'd like to practicing is wrong.
7.) Don't know what you're asking. I think you heard "The cat takes it." probably while smiling. This is very idiomatic. It's hard to explain fully but it could probably be explained as "The cat takes the prize [for being cute]." Though this isn't necessarily the etymology.