Bona Mega is an awesome product and super easy to use. You really can't go wrong with this stuff. They have great tech support as well.
The nails you can drive in and then use a filler. I blend some filler with fine sawdust cut from the same type of wood to match it better. You won't notice the filler but the nails have got to go. It's really not even that hard to pull some wood from the closet, replace the boards and then get new wood for the closet were you won't notice the mismatched pieces as much.
Replace the doors, reuse the hardware. Few hours max. Got to the lumber yard and get blanks. Use a saw track to cut. Easy to make.
Free Circular Saw Cutting Guide Plan - Free Panel Saw Plans
Cut with a sacrificial piece of luan on the bottom to keep from tearing out. Tape both sides of the cut - that helps too and use a new, fine tooth trim blade.
You can rent a hard wood floor sander from Home Depot that is basically 4 or 5 DA (dual action) sanders hooked together with a dust collection sander. Much easier to use for the do it yourselfer that the traditional and hard to learn use floor sanders. Will take longer but hard to mess it up and no sanding marks for going to slow or too fast.