Randomized functions cannot increase statistical distance
Second question: 

$\Delta$ is defined on distributions so the distributions of $f(X)$ and $f(Y)$ are the same, since $X,Y$ are both uniform. The supremum definition makes this clearer.

First question:

Unless the map is one to one, hence a bijective,  when equality holds, the probability that $f(X)=f(Y)$ can only increase compared to the probability that $X=Y$. Prove this for binary distributions and then subpartition and use induction to conclude the general case by using conditional probabilities.