Why is cohomology the direct product of the $H^n$?

This is not a full answer, but maybe gives a hint to the answer of question 1.

An important characteristic class is the Chern character. For (complex) line bundles it is defined as the the formal exponential of the first chern class, i.e. $ch(L)=e^{c_1(L)}:=1+c_1(L)+\frac{c_1(L)^2}{2!}+\ldots$. For higher dimensional vector bundles one defines the chern character by a formal splitting in chern roots. The chern character relates $K$-theory and cohomology, as the exponential has the wonderful property of turning sums into products.

If a cell complex is not finite, the sum, hence the chern character, naturally lives in $\prod_n H^{n}(X;\mathbb{Q})$ and not $\oplus_n H^{n}(X;\mathbb{Q})$. This occurs for the most important chern character of all, the chern character of the tautological bundle over $\mathbb{CP}^\infty$. Note that $\mathbb{CP}^\infty$ is the classifying space of $U(1)$.