The resulting groupoid is equivalent to the disjoint union of groupoids B(H_1(X_i)) taken over all connected components X_i of X. This answers both 1 and 2 in the positive.
To see this, observe that maps in both directions can be constructed using the corresponding universal properties. The fundamental group maps to the first homology group via the Hurewicz homomorphism. Vice versa, any 1-cycle can be converted to a continuous loop (in a nonunique way), and any two such choices will differ by a 2-boundary, which yields the desired map to the quotient under consideration.