In answer to your first question, you might want to look up the Gromov--Grigorchuk topology on the space of marked groups, which topologizes the set of 'marked' groups of rank $r$, where a marking is a choice of generating set of size $r$. References can be found in this paper of Champetier and Guirardel.
It's very well known that there are uncountably many groups of rank two. To see this, one constructs a family of groups of rank two with uncountably many different isomorphism types of $H_2$.
Yves de Cornulier, who is active on MO, probably knows as much as anyone about is an expert on the subspace of metabelian groups in the space of marked groups.
