I think you'll be ok painting the MDF as long as you seal it with a good quality primer first. Primer is quick drying so it's not going to swell the MDF. The smoothness of the desired finish would be the limiting factor on the final coat, imo. Obviously a spray finish will give you the smoothest result and also you can lay down very thin coats, but a roller can give you nice results too, just don't put down too much paint in one shot(maybe use a shorter pile roller). I think it would make for a better and easier paint job if you could remove all the drivers beforehand, just being carefull not to get drips of paint into the mounting flange area.
Using a roller on the grills probably isn't adviseable due to the amount of paint it puts down. I haven't seen these paintable grills before, what are they made of? anyhow, there are plenty of waterbased spray paints available out there, so they should be fine. Good Luck.
