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, and the related computations one can make, I don't know of any other criteria.