One possible condition is that f is non-zero in some neighborhood of a, since in that case, you can write and apply the quotient rule.
I'm not sure about the second question, sorry.
A question asks "If f*g and f are differentiable at a, what conditions on f imply that g is differentiable at a?"
I am also wondering: if f and g are both NOT differentiable at a, but f*g is differentiable at a, what conditions must apply to f and g? I can think of examples but not a general situation; I am also not sure there is even a "general situation"..