The answer to both questions is yes. It's a special case of the fact that, for any colored operad $P$, the category of $P$-algebras is bicomplete (when $V$ is) and is presentable (when $V$ is). For the former statement, you can check out Schwede-Shipley's "Algebras and modules in monoidal model categories", where this is handled in the case of a monad with a condition on the forgetful functor (one that's always satisfied for algebras over a colored operad). For the second statement, I learned this from Lurie's book "Higher Algebra". He discusses it for commutative monoids, and I believe he generalizes that later for other operads. If not, the generalization would be easy.
The above is if you have fixed a color set. If you haven't, then the relevant papers to check out are "A Model Structure for Enriched Coloured Operads" by Caviglia, and " Dwyer-Kan Homotopy Theory of Algebras over Operadic Collections" by Yau.