1. Ideally, you want to do the whole set, with each interval at the same power, finishing while feeling like maybe you could do one more, maybe not. That takes practice with each interval protocol, to get the feel for what the first one should feel like. I like to do them as hill repeats. Then I can judge my relative power easily by how far I get up the hill when the time runs out. If I'm supposed to do a set of 6 X 6' f.i. and on #5 don't get as far up the hill as on the first 4, then I did them a hair too hard and my power has thus dropped off. So I stop with that one, because the training effect of doing them at a lower power is less and I'm better off saving my energy for another day.

2. Each interval protocol targets energy systems is a slightly different way. Some protocols increase endurance at a specific power level, some increase maximum power output, some increase anaerobic capacity, etc. My goals have always been for endurance rides. I'm not interested in anything shorter than a metric century because I'm older so my VO2max is squat but my endurance isn't bad. I don't race formally, but every long event ride is really a race for me. Anyway, the training regimen that has worked best for me is to periodize my workouts. I start with base work. Next is tempo work, especially low cadence tempo work. Then long sweet spot (Z4) intervals, 15'-30'. Then long VO2max of 6'-8'. Then shorter 3' VO2max. Then speed work with 30"-2' efforts. Then full sprints every 5'. Then event taper. Each interval protocol builds on strengths developed by the previous protocol. And of course I mix them together, not usually doing only one protocol/week.

3. Short efforts are called speed work. They are designed to increase your average speed on the flat. They are done at very high power and high cadence. No, they don't just increase your momentary power. Short bursts on hills increase leg strength and ability to put more power on the pavement. Hopefully you'll be able to translate that to greater sustainable power on steep hills. Maybe your legs won't pack it in on 20% grades anymore.

4. Decreasing the rest time is designed to increase the protocol's stress on clearing (burning) lactate. The more lactate you accumulate, the better you get at burning it. That can be a key improvement, i.e medicine one can take as necessary. Longer rest time increases the power one can put down, which has a slightly different effect.

5. "Traditional" periodized interval training uses a variety of protocols, which I talked about in #2. However there is very good evidence that all you really need to do is one set of 4 X 8' X 8 anaerobic intervals once a week. That seems to produce a very good result. Some studies show it producing a higher FTP than the usual long Z4 FTP intervals.

Joe Friel's blog has good info on intervals:
Joe Friel - Intervals, Part 5