As the OP notes in a comment, (2) is clear --- finite sets of point are precisely the zero-dimensional varieties.
As for (1), if $f: X \to Y$ is a dominant (e.g. surjective) morphism of varieties, then the dimension of a
generic fibre is equal to $\dim X - \dim Y$. Now in the case when $f$ is a morphism of group varieties, all the fibres are of the same dimension (in fact they are all isomorphic, since they are cosets of the kernel, and thus all translates of one another). Thus all the fibres are of dimension $\dim X - \dim Y$, and in particular the kernel is of this dimension.
[Note that any (not necessarily surjective) homomorphism $f$ of abelian varieties is a morphism between projective schemes, hence is proper, and hence its image is closed, connected (since the source is connected), and thus is itself an abelian variety. Thus the above reasoning holds, when applied to the induced map $X \to im(f)$.]