First, as others have pointed out, the group of volume preserving diffeomorphisms will be infinite dimensional.
For the second question, there is a beautiful technique known as Moser's trick which answers it. Moser's trick, in fancy language, says that if (M,w) and (M,w') are two symplectic structures on the same manifold, and if [w] = [w'] in H^2(M;R) (de Rham cohomology), then there is a family of diffeomorphism f_t:M->M with f_0 = Id and such that f_1 pulls w' back to w.
For a 2-dimensional compact, oriented manifold (like the sphere), we have H^2(M;R) = R (the real numbers), and a symplectic form is nothing but a nonzero element in R (which can be interpreted as the total volume). Since in this setting, [w] = [w'] iff they both give the same signed volume, it follows from Moser's trick that the group of (signed) volume preserving maps is connected.
If we consider unsigned volume, there will be 2 components to the group diffeomorphisms preserving the unsigned volume. This is because, as others have pointed out, one has the notion of "degree" which shows the map x-> -x is not homotopic to Id, even through just continuous maps (not neccesarily volume preserving). This shows there are AT LEAST two componenets. There are at most two components because every volume preserving diffeo can be connected to Id or (x -> -x) by Moser's trick again.
Edit: I misspoke a little bit. Moser's trick says that if you have a family w_t of symplectic forms, then there is a family of diffeomorphisms as I described above. It's not clear to me that what I said (that it's enough to have [w] = [w'] in H^2) is enough to guarantee that there is a family of symplectic forms connecting them. Further, it seems that Moser's trick only guarantees you have a path of diffeos which starts and ends at a volume preserving diffeo, but may not preserve volume for all time.
However, in the case of S^2 (or any closed, orientable 2-manifold), I can patch things up. Given w and w', volume forms, with [w] = [w'] (i.e, they have the same volume), then the form w_t = tw + (1-t)w' is a path of symplectic forms which connects them. For a fixed t, the form w_t is closed since it's a sum of closed forms (or, even easier, because it has top degree), and is nondegenerate because it's a volume form (integration shows the volume given is that of w). The fact that the volume is constant for each w_t implies that the path of diffeos preserves volume for all time.
(I wasn't able to immediately convince myself that in general, the convex sum of symplectic forms was nondegenerate, hence my initial hesitation. In fact, I think that it need not be nondegenerate.)