Is there a universal countable group? (a countable group containing every countable group as a subgroup)

There isn't a countable group which contains a copy of every countable group as a subgroup. This follows from the fact that there are uncountably many 2-generator groups up to isomorphism.

The first example of such a family was discovered by B.H. Neumann. A clear account of his construction can be found in de la Harpe's book on geometric group theory.

No. There are uncountably many isomorphism classes of finitely generated groups, but a countable group contains only countably many finitely generated subgroups. There is a finitely presented group that contains all recursively presented groups, though.

I don't know how to prove that there are uncountably many isomorphism classes of finitely generated groups. I might try taking a finitely generated group that is not finitely presented and try the groups with the same generators and all subsets of the relations. If the example is too symmetric, like the lamplighter group, this probably won't work, but if you impose random relations of rapidly increasing length, it probably does work.

It is, however, easy to see that there is a universal countable Abelian group.