Proving/Disproving that language L is non-regular/CFL
In fact it turns out that there is no simple method to prove things like this. Except when you want to disprove a statement, then a well-chosen counter example suffices. But also here there are no guidelines how to find them. Experience and trial and error, I would say. In your three questions I see three different approaches.

For (1) go look for a counter-example. Since the statement is true for most languages, you have to look for trivial special cases, but still contradicting the general claim. 

For (2) you can try contra-positive. Instead of $A \Rightarrow B$ consider the equivalent $\lnot B \Rightarrow \lnot A$. This is then a (somewhat exotic) closure property. Can you recover $L$ from $L'$?

And I am afraid that (3) looks like simple Boolean set algebra. It is in fact the most straightforward of the three questions.