I am mostly interested in the case when you have a smooth degree $d$ algebraic surface $X$ over $\mathbb C$ and we can define three distinct groups: $\pi_0(\mathrm{Diff}^+(X))$, $\pi_0(\mathrm{Homeo}^+(X))$ which is isomorphic to $O(H_2(X;\mathbb Z),\langle\cdot,\cdot\rangle)$, and $\pi_0(\mathrm{Symp}(X))$.
Question: What is known about each of these groups? Are they ever finitely generated (for diffeo/symplectomorphisms)?
What about the relative mapping class group case, i.e. if $\Sigma\subset X$ is a smooth algebraic curve, then what is known about $\pi_0$ of the group of homeo/diffeo/symplectomorphisms which fix $\Sigma$ setwise?