(Re: 2 & 3)
Since $\operatorname{Sym}^n\mathbb CP^1=\mathbb CP^n$ («by Viète's formulas»), $\mathbb CP^\infty$ is the free abelian monoid generated by $\mathbb CP^1=S^2$ (with some fixed point as a unit). (This is exatly the operation «representing» tensor product of $U(1)$-bundles aka addition in $H^2$ on $[-,\mathbb CP^\infty]=\operatorname{Bun}_{U(1)}(-)=H^2(-;\mathbb Z)$ — cf. other answers.)
Unfortunately, this operation lacks (stict) inverse. But by Dold-Thom theorem $\widetilde{\mathbb Z[S^2]}:=\mathbb Z[S^2]/\mathbb Z[pt]$ (the free abelian group generated by $S^2$ with some (fixed) point as a unit) has homotopy type of $K(\mathbb Z,2)$. Moreover, the natural map $\mathbb CP^\infty=\operatorname{Sym}^\infty(S^2)\to\widetilde{\mathbb Z[S^2]}$ is a homotopy equivalence.
(And any $K(G,n)$ can be made an abelian topological group in analogous way: $\widetilde{G[S^n]}$ has desired homotopy type.)
