I suggest the following lecture notes of Bruhat:
www.math.tifr.res.in/~publ/ln/tifr14.pdf
Chapter 3 & 4 should answer most of your questions.
For example, there are statements like this :
Proposition 1(pg.19). To every analytic representation h : G −→ G′ there corresponds a map dh : U(G) → U(G′) which is a representation of algebras such that ( f ◦ h) = (dh() f ) ◦ h.
Corollary (pg.36). Let G and H be two Lie groups having g and J as their Lie algebras. If G is connected and simply connected, to every representation π of g in J, there corresponds one and only one representation f of G → H such that d f = π.
If you are interested in semisimple, connected and simply connected groups only, both give you an isomorphism between the categories. Equivalence of categories is weaker then isomorphism. Moreover, the categories of representations are both semisimple in this case, i.e., reps decompose into irreducible ones. Thus, also your second and third question can be answered affirmative.