A lot of this depends on the scale of the project and the size of the crew / facilities being used. Here, we generally have 3 or 4 people doing pre mixes. One doing backgrounds, one doing effects, and another doing dialogue. Sometimes we'll have a 4th doing sound design / special fx stuff. Generally the music gets to us mixed, or at least in pre-mix stems, so we usually don't touch it until re-recording.
Once all of those parts are done, they get mixed down into stems and send to our re-recording mixers. We usually use 2. One handles Dialogue and rides the music stem, the other does backgrounds and effects. If music is broken out into a larger mix instead of a stem, we'll have a music mixer too.
Generally what happens is that one guy will be working on his pass for a section (let's say the dialogue mixer is doing his thing) and he controls the transport and all. While this is going on, the fx mixer might be getting general levels, but won't really be doing much fine tuning. When the dialogue mixer is done his section, he'll print it and then the fx mixer will do his fine tuning. We usually operate like this, back and forth for a while, until the project is done.
Things might speed up and we might try to get FX and DX mixing at the same time if we're really in a crunch.
As far as order, we really don't do things in a specific order really, but we do try to make sure dialogue gets the main focus. It often doesn't get mixed first though.
Does this help at all? Let me know if you want me to expand on anything