It is well known and not hard to prove that $\binom{pA}{pB}\equiv\binom{A}{B}\mod p$ where $p$ is a prime. Now, how can we extend to show that this congruence holds $\mod p^2$.
Finally, can we extend this to $\binom{pA+a}{pB+b}\equiv\binom{A}{B}\binom{a}{b}\mod p$?
You cannot show that it holds modulo $p^2$ because it is wrong (try $A=p$ and $B=1$).
Your second question is just Lucas theorem for $a$ and $b$ at most $p-a$.
See http://en.wikipedia.org/wiki/Lucas%27_theorem