Indeed, the literal negative answer to the first question includes the standard counter-example as given by @JoseCarlosSantos.
Still, modifying the question/example somewhat in various ways gives a more positive answer. First, if we insist on "algebraic" representations (in some sense...) this could be arranged to exclude the counter-example.
Or, for example, instead of $G=GL_2$, trying $G=SL_2$ avoids the counter-example.
In either style of avoiding that (and related) counter-examples, H. Weyl's "unitarian trick" tries to arrange that a finite-dimensional repn (under some hypotheses...) of a reductive (linear...) real Lie group can be complexified, and then restricted to a repn of the compact real form, thus reducing many questions to the repn theory of compact Lie groups.
Thus, under mild (but non-trivial) hypotheses such repns are completely reducible, and so on. But you are right, that they don't have invariant inner products for the (original) non-compact group, although they do have such for the compact real form, which enables Weyl's proofs (under various suitable hypotheses to make the argument possible).
And, yes, then a classification of ("algebraic"?) irreducibles by highest weights succeeds.