I could set it over the course of the seasons but there is no guarantee we can shoot the scenes in order.
Movies are
never shot in order, for the exact reason of scheduling - you need to work around everyone's schedule, including the location's schedule. Plus, it makes sense to shoot all the scenes at the same location at the same time, for example.
Do you have any good tips on how to get one minute of a fight/chase done in less than 8 hours?
Have rehearsals. If you get your fight choreographed
before you get to set so that the actors know exactly what's going on, then on the day you can just shoot. You shouldn't have to do more than a few takes of each angle, and you shouldn't be getting 8 hours of angles (to me 8 hours is probably 15-20 shots, more if you don't change lighting between; are you really going to use 15-20 shots in one minute of action?). Cut your shot list down, or decied on exactly what shots you need before the day. Or accept that your shoot days will be that long, and deal with it.
Or make shorts until you're comfortable enough with askign people to work weekdays.
I realize I should have done it with less shots, since I am on a budget, so with only a few shots, say 4, do you think I could cover five minutes of dialogue in 2 hours? This way I can devote more of the shoot time, to the scenes that require more shots and are more complex.
You need to come up with a shot list. Some dialogue scenes can be covered in 4 shots, some need to be covered on 12. It all depends on the scene, what's happening, the subtext etc. And then, the 12 shots can be static on sticks, whereas you can have 4 dolly shots, or 2 dolly shots and a steadicam shot, which is going to take as long or perhaps longer than your 12 shots.
Get a good 1st AD, and do comprehensive pre-production. It can be tempting to just jump in to Production, but without spending time making sure you've done all your pre, you'll end up working much slower.