Wood trim and new gutters: Gutters first or paint first?
You shouldn't just try to calculate this based on how long it will take you to paint the fascia one time.  Consider the fact that you will need to paint it and repair parts of it every couple of years forever if you don't get it wrapped.  My advice is to spend the money to get it wrapped.  

To answer your questions more specifically:  Pull the gutters down and paint when the forecast looks nice.  If you get some rain, or even a storm, before the new gutters go up, that's no big deal.  It's constant runoff from your roof that would eventually cause problems.  

Don't forget that your old aluminum gutters may have some value at the local scrapyard.