I'd get some bar tape. Bare metal gets awful slippery when your hands are sweaty.
You can leave the rear wheel just like it is.
I'd have the front brake using a lever on the right. That way, you can use your brake while making traffic signals with your left hand. It also shortens up the brake cable and makes it look neater. I'd keep the rear brake, since it's a singlespeed. Flying downhill without a rear brake can be scary, especially if there are curves.
If you decide to get a track hub, you can probably ditch the rear brake but I'd still keep the lever for symmetry and also riding on the hoods is a good thing. Adjust the brake lever(s) up to where they were in the before picture. Maybe a hair higher, even.
(sorry, missed the part where you wrote you were getting a new wheelset. I'd still keep both brakes but that's just me.)
What sort of riding do plan to do with this bike?