Is APX contained in NP? A problem P is said to be in APX if there exists some constant c > 0 such that a polynomial-time approximation algorithm exists for P with approximation factor 1 + c.

APX contains PTAS (seen by simply picking any constant c > 0) and P.

Is APX in NP? In particular, does the existence of a polynomial-time approximation algorithm for some approximation factor imply that the problem is in NP?
APX is defined as a subset of NPO, so yes, if an optimization problem is in APX then the corresponding decision problem is in NP.

However, if what you're asking is whether an arbitrary problem must be in NP (or NPO) if there is a poly time O(1)-approximation, then the answer is no.  I don't know of any natural problems that serve as a counter-example, but one could define an artificial maximization problem where the objective is the sum of two terms, a large term that is easily optimized in P, and a much smaller term that adds a small amount if part of the solution encodes an answer to some hard problem (outside of NP).  Then you could find, say, a 2-approximation in poly time by concentrating on the easy term, but finding an optimal solution would require solving the hard problem.