What type of water based paint makes the most sense for durability and ease of application? (Acrylic? Latex? Waterborne Enamel? Waterborne alkyds? Something else?)
Yes. Sorry, couldn't resist. The truth is that nearly anything will work for this, from the old-school choice of milk paint to the most modern waterbased formulation. There's no need to overthink here, because this isn't a project in need of high wear resistance or with other specific performance requirement I think you should just pick the product you're most comfortable with in terms of cost and ease of application that gives the surface gloss you're looking for.
In terms of application tools I would highly recommend foam rollers for the large flat expanses, with small synthetic brushes for corners and detail zones. You could use small paint pads in place of the brushes, these can work well but I've had very mixed results in how well they hold up to use and washing so I can't recommend them without reservation.
What do you do with regards to painting these holes? Q-tip dip into paint, then into the hole during the initial priming?
Painting every hole would be a chore, no doubt about it and using a Q-tip to do the job is good OTB thinking on how to do it, but I wouldn't recommend it. Obviously you want to but I think it's sure to be more trouble than it's worth.
Firstly when you paint you reduce the diameter of each hole enough to make for tons of trouble with over-tight fit of the pins, making them hard to insert (even if they have a chamfered leading edge) but sometimes nearly impossible to remove. At least without resorting to pliers.
Even if the fit is not very tight you're sure to have wear or paint lifting inside the holes if and when you move a shelf and take out its corresponding pins, which will leave visible dark brown/black peeking from those holes. Now you could touch those up each time but I presume this is something you'd want to avoid.
So instead of going for your Tedium Award, Class 1 by painting each hole I suggest you get it by doing the opposite, masking off each one as you paint :-) Appropriately sized dowels, slightly tapered towards their tips, are perhaps the ideal mask here, but if you went with bare wood you'd need a large number because you won't easily be able to re-use each dowel after wiping off the wet paint without risk of slight paint transfer. So one thing you might try is making a few specifically for this by sanding your dowel lengths lightly (to reduce their diameter just a smidge) and giving them a couple of coats of shellac or varnish to seal the surface. That way the paint will wipe off much more easily and thoroughly and with luck you'll be able to re-use each one immediately without a problem.