On integer programming Integer programming is NP-hard.

What is the status of integer programming problem that decides between existence of $\leq1$ solution and $&gt;1$ solutions (note $0$ solutions falls in $\leq1$ category)?

Integer programming in fixed parameters is P.

What is the status of integer programming problem in fixed parameters that decides between existence of $\leq1$ solution and $&gt;1$ solutions (note $0$ solutions falls in $\leq1$ category)?
(1) As finding a second satisfying assignment to a 3SAT formula is still $\mathsf{FNP}$-complete (indeed, it is $\mathsf{ASP}$-complete, see Theorem 3.5 of [1]), and we can encode 3SAT as an integer program by a parsimonious reduction, finding a second integer point in an integer program is also $\mathsf{NP}$-hard.

(2) Barvinok [2] showed that in fixed dimension, you can actually count the number of integer points in polynomial time, so the answer to your second question is $\mathsf{P}$.

[1] Takayuki Yato and Takahiro Seta. Complexity and completeness of finding another solution and its application to puzzles. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, E86-A(5):1052–1060, May 2003. (freely available author's copy)

[2] FOCS 1993