Let $f:X \to Y$ be a closed immersion between smooth projective complex varieties. Suppose that the codimension of (the image of) $Y$ in $X$ is equal to $r \ge 1$. This induces the Gysin morphism $ f_{*} : H^{k} (X) \to H^{k+2r} (Y) $.
Is there any known example/criterion for which the Gysin morphism $f_*$ is surjective, at least when $k$ is even?
It seems natural to approach this by trying to compute the mixed Hodge structure on $Y \setminus X$. The cohomology carries a weight filtration $W_\bullet$ and the Gysin map will be surjective if and only if the lowest weight part $ W_{k+2r}H^{k+2r}(Y \setminus X, \mathbf Q) $ vanishes.
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: $$ \cdots \to H^{k}(Y)\to H^{k+2r}(X)\to H^{k+2r}(U)\to H^{k+1}(Y)\to\cdots $$ where $U=X-Y$ is the open complement of $Y$. Apart from that, and the related computations one can make, I don't know of any other criteria.
I suppose that you are discussing Betti cohomology with coefficient in $\mathbb{Q}$.
Using the long exact sequence $$\cdots \to H^{k}(X,\mathbb{Q}) = H^{k+2r}(Y,Y \backslash X,\mathbb{Q}) \to H^{k+2r}(Y,\mathbb{Q}) \to H^{k+2r}(Y\backslash X,\mathbb{Q}) \to \cdots$$ we see that the surjectivity of the Gysin morphism $f_\*$ is equivalent to the vanishing of the pullback morphism $i^*:H^{k+2r}(Y,\mathbb{Q}) \to H^{k+2r}(Y\backslash X,\mathbb{Q})$, where $i$ is the open immersion $Y\backslash X \to Y$. (Since your $Y$ is smooth and projective, the Hodge structure on $H^{k+2r}(Y,\mathbb{Q})$ is pure, which explains the equivalence given by Dan)
We know that $Ker \ i^*$ is a sub-Hodge structure of (Hodge) coniveau $\ge r$ of $H^{k+2r}(Y,\mathbb{Q})$, it says in particular that if you want your Gysin morphism $f_{*}$ to be surjective, your $H^{k+2r}(Y, \mathbb{Q})$ should at least have coniveau $\ge r$! ($\leftarrow$ This should be seen as an exclamation mark, not a factorial notation...)
