Inverse Function: unique? Is it true in general that the inverse of a function is unique if it exists? Why is this so?
Clearly inverses in groups are unique. However, that seems not directly applicable in this  case...

If it is a two-sided inverse, then yes. To see why assume that $f_1,f_2$ are both
two-sided inverses of $h$. Then
$$f_1 = f_1\circ (h\circ f_2) = (f_1\circ h)\circ f_2 = f_2$$
Note that this proof implies that if $h$ has a left inverse and a right inverse, then they are necessarily the same. 
