So, taking the first of your questions:
"How many bit strings of length 10 have exactly three 0s?"
is equivalent to asking: in how many ways can I make a set of 3 positions out of 10? (the positions being the positions in the bit string where the 0s occur - the other 7 positions then have 1s).
Can you answer that question?
Hint: Combinations; Binomium of Newton.