yup- another vote for remove everything that is removable from the frame. You can better sand down/strip the old paint this way, and it also allows you to properly prep the frame. You will want to remove all rust, so having the BB threads exposed is good.
Hours of prep make for easier and better painting- regardless of if you do the painting or pay someone. ALL rust should be removed with emory cloth and/or a chemical bath. If you paint it, use something that preps the steel(primer, etcher, etc. there are multiple options) and then apply the paint over the prep'd frame. It will look better and be more consistent.