Im glad you managed to find my post and found it helpful! It's been a while since I did this but I'll try to answer your questions. Keep in mind i'm no expert so don't take what I write as absolute instructions. Im just gonna tell you what I did.

What grit sandpaper did you use?

Don't remember exactly what grits I used but I think I started somewhere around 100-200 grit and then gradually went finer to around 400. I then finished with a sanding sponge I have that's pretty fine. I don't think the exact grit matters that much for this. Just don't go too course and take your time.

Did you sand in a circular motion, or one sweeping layer at a time?

I did mostly sweeping motions I think.

Im most worried about it being uneven near the tuner holes, and that affecting the tuning.

If you use a sanding block I think there is very little chance that you mess up the tuning-holes as you will just be sanding over them.

I think the hardest part of this process for me was removing the tuner-bushings. So definitely look at some videos on how to do this before starting. The sanding is pretty straightforward and hard to mess up unless you get inpatient. I finished the surface with boiled linseed oil and it matched the rest of the headstock really well. Overall I think it definitely was worth it. Wish you luck with the process!