Can Lenstra's algorithm output all feasible solutions in O^*(f(k)) time where k is the number of variables and f is a computable function in k?
No. The number of feasible solutions cannot be upper bounded by $f(k)n^{O(1)}$.

Consider the integer program
$I_n: 1 \le x\le 2^n$
with the integer variable $x$. So, $k=1$ and the program can be described with $O(n)$ bits. But it has $2^n$ solutions, which is exponential in the instance size.