Is the Picard Group countable? Is the Picard group of a (smooth, projective) variety always countable? 
This seems likely but I have no idea if it's true. 
If so, is the Picard group necessarily finitely generated?

No. Even for curves there is an entire variety which parametrizes $Pic^0$(X). It is called the jacobian variety. The jacobian is g dimensional (where g is the genus of the curve), so in particular if g > 0 and you are working over an uncountable field the picard group will be uncountable.
