The propagation is an analytic process that affects all frames, including the base frames you paint on. It simply detects patterns and edges based on pixel motion analysis forward and backward. By brushing over regions you merely define a reference in terms of the amount of motion, size an evenness of the patterns and that sort of thing. It's not a process where you select absolute pixels or anything of the sort.
what exactly is going on each time i make an edit and then press spacebar to let the rodoscoping program run
actually you don't use the spacebar or the Numpad 0 to make Rotobrush run (it's called propagate frames). it does this automatically based on where your CTI is. if you move the CTI forward or backwards from the initial base frame, it will cache the frames until that frame. you should move frame by frame and it will follow along. if you press space bar or Numpad 0 you are caching the playback to Ram but it could include your propagated frames or not - like in a case like this:
here there are frames loaded into Cache due to Spacebar or Numpad 0, but some of them do not include propagated frames... all this to say is that you should first notice the roto brush propagation cache and focus on that. later you can preview it and load it to playback in real time. the full recommended workflow is the link at the bottom of this post.
is the program going back to my earlier frames and making edits?
No. propagation of frames occurs away from your base frame to where your CTI is in the direction the arrows are pointing. if you make a change it will not edit previous frames i.e against the direction of the arrows, but only affect how your selection will be modified in subsequent frames (away from your CTI and base frame) that's why it's recommended not to advance to many frames away from your CTI and then go back and make corrections. you need to "teach" rotobrush to work so you make a brush correction and as you move according to the arrows your make "Corrective strokes"
if i make a change with the refine edge tool, do i need to hit spacebar so the program reruns all the frames to get the updated results or does it automatically do that?
same thing like the roto brush tool.
very useful and comprehensive information can be found here: Roto Brush and Refine Matte in After Effects make sure you read it all the way through. Rotobrush is magic but you have to use it carefully and also know when not to use it at all.
here's a freebie: Adjusting propagation settings by Chris Meyer on Lynda. I recommend taking the whole course.