There were quite a few different questions, so forgive me if my answer is somewhat fragmented.
The Néron-Severi group (divisors modulo algebraic equivalence) is finitely generated over any field for any non-singular projective variety $X$, this is Severi's theorem of the base.
In the case of curves, it is always isomorphic to $\mathbb{Z}$, with the isomorphism given by the degree map.
There is a natural morphism $NS(X) \to H^2(X,\mathbb{Z})$, which is injective on the free part of the group. This will then give you an upper bound for the Picard number.
Torsion in $NS(X)$ naturally lives inside $H^1(X,\mathbb{Z})$, so if this is torsion free then so is $NS(X)$. Also, torsion divisors can never be ample because they are numerically trivial, and ampleness is preserved under numerical equivalence.