How good do I need to make the primer coat

It depends ;)

I don't tend to do a final sand for the very reason you mention. However there have been times when the finish hasn't been to my liking so I have sanded - but in these cases I use the finest grade paper and sand very lightly to just take off the drips and/or dust flecks that have settled on the wood.

So if the penultimate coat is good you might be OK without sanding, but sanding does provide something for the final coat to bind to if the previous coat was applied some time ago, so ultimately it will provide a more durable finish. If you are painting the second coat as soon as the first coat is dry then you don't need to bother.

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.