Yes, you should remove BB and headset cups too. Then you need to find a local powder coater, or if you get a deal from the auto guy, take him up on it. The DIY option takes a lot of skill to do right. Better to hand if off. "I know nothing about painting such things" = hand it off.
Bottom brackets take routine maintenance, no matter how "smooth" it feels, grease gets OLD and no longer lubricates it properly. Then you do damage to the BB.
Powder coat is super durable and is usually cheaper than wet paint, and no more expensive than the stuff you will need to DIY.
Do yourself a favor, and list your LOCATION in your profile. Who knows, there may be someone local to you that can assist!!