Mainly to stop moisture getting trapped in the paint finish, though too cold and the paint doesn't dry as well. If you use cans or a spray gun, the gas expansion cools the paint and solvent evaporation cools it more. Moisture in the air condenses out and you end up with a milky finish. Sometimes this will clear, and sometimes it won't. So you really want 18°-22°C ambient for optimum results. Also helps with spray cans to warm them in warm water first to get the paint moving freely.

Grain filling depends on the wood. Closed-pore woods (like your basswood body) don't need grain filling, but open pored woods (like ash or mahogany do). No need to go too mad on fine grits before painting. You basically want to remove any tooling marks and get the surfaces of the body and edges as flat as possible. Don't be afraid to start off with a coarse paper, say P60, then move to P120 and P180. Flat sanding pads are vital here. You could go to P240, but not too smooth in order to help the paint stick to the wood.

Going with a solid colour? Then a primer/filler on first is a good idea. Only dry sand the primer as it's not normally that waterproof.

You've got the option of painting the neck and body separately (physically easier) or glueing the neck on and spraying it as one. Pre-glueing allows you to fill in gaps around the neck join area before priming. You then end up with a nice smooth neck/body transition. If you glue the neck on after finishing, you've then got to fill the gaps (if you feel the need) and re-do all the finish in that area.