Drywall – Prime/paint drywall ceiling in bathroom around ceiling light

Personally I would pull them down and drywall the entire thing. The edge of cabinets is going to not look very well when your done if you don't and will stick out like a sore thumb. It's easy enough to pull down the cabinets too.

Also I would remove the drywall from the edge by the wall too and mud the corner. It's going to make it cleaner looking in the end.

Ideally, if it's standard sheetrock, I'd suggest removing it. You normally want to use 'greenboard' (sheetrock with a waxed paper layer) or, even better, the paperless wallboard products like Densarmor (they use a fiberglass layer rather than paper). The paper is what the mildew will like to 'eat' so removing as much of that as you can, the better.

Barring that, they do make paints specifically for bathrooms. They usually have some sort of anti-mildew additive.

Finally, be sure the bathroom is properly vented. That, alone, is the single best solution. If there is no ceiling fan, invest in that first.

To answer the specifics:

Latex is water based, so that's the same thing. And is what you want to use--specifically ones labeled for bathrooms and high-moisture areas (ie, they have the mildew preventative additive). You COULD use Oil based paint, and that has historically been something to use in a bathroom, but latex paints have come a long way and Oil based interior paint is now a rarity and likely not what most DIYers would want to use anyways (it's a bit more finicky and smelly to apply and a lot more messy to clean up).