It seems you get class rank for anything you do whilst you are playing as that class. Though if you spawn into a vehicle you will be getting rank for that vehicle class like pilot, tanker or cavalry. If somebody else is driving the vehicle then it should be fine to spawn into it.
To get Support points you can do the following:
Give out ammo
Capture flags (Earns you a lot of points)
Everybody likes ammo, throw it all over the place. When you see somebody who seems like they are to far away to give ammo sometimes you get the popup to press
to give them ammo. Do this and you will throw the ammo pack (not crate) straight to them even if they are very far away. The key to getting this quick is to get points for playing the objective. So capture many flags or destroy the telegraph posts.
Note: I edited my solution as it was initially about how to get around the glitch but that has been patched.