I guess that the following are true; maybe classical? Is there a reference?

1. Let $X, Y, Z$ be connected pointed CW-complexes, $f:X\to Y$ and $g:X\to Z$ pointed maps. Assume that for every $k\ge 1$, the kernel of $f_*:\pi_k(X)\to\pi_k(Y)$ is contained in the kernel of $g_*:\pi_k(X)\to\pi_k(Z)$. Does it follow that there is a pointed map $h:Y\to Z$ such that $h\circ f$ is homotopic to $g$?

2. Let $Diff(M)$ be the group of the smooth diffeomorphisms of a closed separated manifold $M$, with the smooth topology. Does $Diff(M)$ have the homotopy type of a numerable CW-complex? Are its integral homology groups numerable?