Definitely dry fit your parts together to see whether or not you'll have large gaps you need to fill. Filling after you've started painting is not easy and something I try to avoid. However, sometimes you just can't. If the fill won't be easily visible or can be disguised somehow, then it's not as critical. I find using
clay shapers can be helpful in smoothing out a full transition on a painted piece. Here's one I did on a current project. The cuff did not fit onto the arm as well as I had hoped. It left a gap which is circled in the center image. I was able to fill it with putty, smooth out the transition, and then repaint the end of the arm. Again, this isn't ideal, but you can make it work if you have to.
It also doesn't hurt to use the black variety of milliput, so if you miss a section (or can't reach it to paint over), it will not stand out as much.
As for actually painting the parts separately, I try to only do as much as I have to. Often that means just giving them a dark base coat and perhaps a little bit of highlighting. It's the sides that won't be accessible after it's assembled that I care about. Often these won't be easily seen and will often be in shadow anyway, so a dark base coat is enough. I need to know the final angle of the piece to get the highlights and shadows to look right, so painting it off of the main figure is a challenge.
Another challenge for painting parts separately is how to hold them while you paint. If you're pinning the piece on (which is generally a good idea), you can use that and a pin vice to hold it. I find this technique works well for banners too as you can just slide the pole down into the vice.
I've got a universal jeweler's vice that I often use to hold the figures while I paint. I just clamp is on to the pins in their feet.
But using the little pegs, you can grip off shaped parts as well. This can be useful for shields as you can see below.
And for another project I'm currently working on, I just used some sticky tack and stuck the extra bits to a piece of cork while I was painting them.