The only problem I can potentially see is that the off white paint on the garage door may be a factory applied oil based paint or a powder coating, in which case you may have trouble getting the grey paint to stick well to it.
Here's how you test for paint adhesion: Apply a little of the grey paint to the off white coating; maybe a 2 inch by 2 inch square. Allow the grey paint to dry, and then score the grey square both vertically and horizontally at 1/4 (approx) intervals with a razor to make a "checker board" pattern on the paint. The idea is to cut through the grey paint film, but not the off white paint. Now, apply 2 inch wide ordinary masking tape over the checkerboard pattern, press the tape down tight and pull it off quickly. If more than 80 percent of the grey squares stay stuck on the off white paint, then you're getting good paint adhesion, and you don't need to worry about sanding the off white coating down to improve the adhesion of the grey paint.
If you're getting good paint adhesion, I would use a brush in those areas where you might have difficulty getting a roller to go, and then go over (as best you can) the brushed area with a "mini paint roller" (as seen below) to eliminate any brush marks.
The above image really doesn't give you a good idea of how large these mini rollers are. Typically, they're about 4 inches wide and about an inch in diameter.
The advantage of using a mini paint roller is that the ends of the rollers have pile on them, so you can paint both sides of a corner at the same time. I'm thinking that would be handy when painting the joints between the garage door segments. Phone around to the paint stores in your area, and any store that sells "Dynamic" painting products will carry these mini rollers and the frames (and trays) for them. You'll need the special frame, but any paint roller tray will do.
Our garage door has weathered extremely well no blemishes or chalking or powdering...
So far as I know, chaulking and powdering of exterior paint is the same thing. It's most commonly referred to as "chaulking" because wiping your hand over the chaulked paint will leave a white powder on your hand the same as wiping it over a school chaulk board would.