Does new drywall need 2 coats of primer

When you say you can see the brush marks, do you mean that they're actually irregularly surfaced? I mean - are we talking just a visual effect or an actual difference in the depth of the paint?

If the latter, do a skim coat with lightweight joint compound and when it dries, either lightly sand OR smooth with a large, slightly damp sponge, then apply primer.

As for your own brush strokes - orient the roller so the roller bar leads your motion. If you're moving right, put the bar on the right. If you're moving left, put the bar on the left. The most pressure will be on the roller bar side, so keeping that side on the "front" of your motion will mean the lighter touch side (the back) will smooth out the uneven strokes as you pass over it again.

Now if its not a real issue with texture, just that your seams are obvious - then just prime it again with a good HEAVY primer, using the proper brush techniques, and you'll be fine. :)

Yes, you should prime the whole surface.

Sealing/priming helps prevent penetration of vapors and moisture from the air into the drywall. If you don't seal the entire surface, even though it's behind cabinets, penetration can happen - mind you it's not going to be significant but it can happen.

Plus the time involved to paint behind those cabinets that aren't there yet is minuscule.

This is unrelated but I don't understand your painter's objection. Painting a wall with no cabinets on it is orders of magnitude easier than painting a wall with cabinets already mounted. With cabinets in place, you've got to tape edges, you risk getting paint on your new cabinets, you've got a much harder time accessing the exposed wall (cabinets get in the way), and so forth.