Since you accepted the answer on Operations Research, you should close it here, and on all of the other sites you cross-posted it on, or explain clearly how the answers you are hoping to get on these sites would differ from the one on OR.

I don’t know for sure, but I sort of doubt that’s possible.

The Taylor series question is interesting but should probably be asked as a separate question. It's best to ask multiple questions each with their own (narrow) scope rather than to have multi-part questions.

It's not clear to me exactly what you are trying to do, then. Are you saying you want a linear reformulation of $F(X)$ that is exactly equal to the original formulation of $F(X)$ when $X$ is relaxed to be continuous?

Relax $X$ how? More importantly, why? It seems you are worried about the NP-hardness of your ILP. But CPLEX and Gurobi eat NP-hard problems for breakfast; you shouldn't over-think it. Have you tried just giving your ILP to CPLEX or Gurobi? They will do the relaxations for you within the branch-and-bound (or -cut) scheme; you don't need to do that part explicitly.

I think the last sentence should be "...with coefficient $c_*$."

What about an LP with an infinite number of constraints? The feasible region can still be bounded, but there may be no optimal or lexicographically smallest solution.

Optimization over unions of polytopes is called disjunctive programming -- here's a reference: onlinelibrary.wiley.com/doi/pdf/10.1002/9780470400531.eorms0‌​262