Whenever I read a textbook or get help with a proof, I'm always suggested a very obscure and tricky step which usually helps me immediately solve the rest of the proof because everything else seems to becomes obvious after that one single trick. I believe this isn't helping me because it's just that one step I cannot think of, and I have no idea how other people are thinking of that idea. Do they work backwards, recall something from memory that is very similar to the problem, just trial-and-error various ideas very quickly in their head, etc.?
As a short example, page 7 on Rudin's "Principles of Mathematical Analysis," 3rd edition, on the proof for proposition 1.16(b) which says that if $x \neq 0$ and $y \neq 0$, then $xy \neq 0$. I would not have thought to try and use $1 = (\frac{1}{y})(\frac{1}{x})xy$ and then use contraction to show it can't be true. How would you know to use contradiction in this case? Is it something to recognize? And after you determine this, how do you know to use something like $1 = (\frac{1}{y})(\frac{1}{x})xy$? Or maybe it's the other way around; trying to prove something and accidentally finding a contradiction? I don't think I would have thought of this unless I spent considerable time, particularly on such an "easy" (I assume easy because it's on page 7 and pretty elementary) proof.
I know it's very subjective/situational, but I sometimes wish authors of texts and answers on this site would also include the thinking steps on how they arrived or thought of the lines in their proofs instead of just showing all the detailed work without the explanations.
Check out How to Prove It--it sounds like you'd benefit a lot from it (in fact, I wish I'd bought it sooner).
All those questions you have? That book will answer all of them--it teaches you the basic (yet fundamental) techniques of proof-writing via lots of worked-out examples and practice problems.