Can all packing/covering problems be rephrased as set packing/covering problems?
If we restrict ourselves to problems in NP, then yes. There are NP-complete problems of either category, so (many-one poly-time) reductions exist; we can view these as "rephrasing".

Since integer programming is also NP-complete, these same is true for that one.

Problems outside of NP do not reduce to integer programming, but I am sure that any (meaningful) complexity class contains examples of both covering and packing problems.

It's not the "type" of a problem that determines its hardness. There are satisfiability problems all the way from linear complexity to undecidability, to name just one example.