Integer Problems are NP-HARD
No, it doesn't mean that.

The set of NP-hard problems is upward-closed in the sense that there is no upper bound on the hardness of problems in it. You are correct to say that the class of NP-hard problems contains undecidable problems.

That does not imply, however, that every NP-hard problem has "unsolvable instances".


There is no such thing as an "unsolvable instance" in integer programming. There always is an answer, true or false (considering the decision version). Every finite set of instances is trivially computable. Only infinite sets can be undecidable.
We do know upper bounds on integer programming. For the decision version, we know that it's in NP; the optimization version is in NPO. Therefore, we know IP is decidable.