Yes, homology is a coalgebra, with the coproduct given by the pushforward X -> X x X. This is dual to the ring structure on cohomology, so at least over a field, they determine each other.

Yes, homology is a coalgebra, at least with coefficients over a field, with the coproduct given by the pushforward X -> X x X. This is dual to the ring structure on cohomology, so they determine each other.