Let g, h be elements of a finite group G and let H be a subgroup of G.
Which are the possibilities for |gH 
hH|?
anything
any number less than or equal to |H|
0 or |H|
No, the size of |gH| is at most |H|, so the size of the
intersection cannot be more than that.
No, two
equivalence classes of the same relation cannot meet in an arbitrary
subset of one of them.
That is correct. Two equivalence classes either coincide or are disjoint.
Note that |H| = |gH| for every g.