Question 1: Is there any known example for which the Gysin morphism $f_∗$ is surjective, at least when $k$ is even?
Yes, consider the immersion $\mathbb{P}^n \to \mathbb{P}^{n+1}$. It is surjective for all $k$ and $n$.
Question 2: Is there any known criteria to know if the Gysin morphism $f_∗$ is surjective?
This is precisely the Gysin long exact sequence: $$ \cdots \to H^{k}(Y)\to H^{k+2r}(X)\to H^{k+2r}(U)\to H^{k+1}(Y)\to\cdots $$ Apart from that I don't know of any other criteria.