Well, here is a little update. I ended up painting the stand over the course of a week, allowing time to dry between each coat:
(1) Two coats of Kilz 2 Latex primer/sealer/stainblocker
(2) One coat of Benjamin Moore White Dove (OC-17) Latex paint
(3) One coat of Minwax Water-based Polycrylic protective finish.
I also bought some soft-close overlay door hinges to attach the doors to the stand. And now I have a somewhat finished product. In the last pic, you can see that I screwed up the placement of the hinges by like 1/16", which made the doors not open properly. So basically, I had to flip the doors around and attach the hinges to the other side. So I have some minor patch work to fill in the hinge holes and pocket holes, but at least the stand is complete enough to move to its final resting spot. Next up.... PLUMBING! Stay tuned.