Rest periods are not specified in the program. It's not that the program doesn't address rest periods. It addresses them by specifically stating that rest periods are not specified ;-). In other words, rest however you want as long as you get all your reps, anything from the minimum you think you need to spacing out the work throughout the day.
I've done a lot of cycles of the fighter pullup 3RM program with weighted pullups. In my experience the key making it work is to not use an all out struggling 3RM. I found that starting with a tough 5RM turned that into a very comfortable set of 5 by the end of the program, which is less than 2 weeks if you don't take extra days off, and makes the program a lot more sustainable in terms of running consecutive cycles with increasing weight, or moving on to the 5RM program.
If currently struggle to get 3 bodyweight reps, I'd do a ladder ramp up program before starting the 3RM program proper, something like:
1,1,1,1,1
1, 2,1,1
1,2,2,1
1,2,2,2
1,2,3,1
1,2,3,2
1,2,3,3
1,2,3,4
Stay on each level for longer than one session if necessary.
I'm not big on warmups for most normal training and I've never really done anything to warm up for pullups. Not that there's anything wrong with warming up (especially for high load or high skill training), and there are drills where I prepare with some minimal separate warmup moves and/or a brief ramp up of the load to my working weight. But I don't have any specific advice about warming up for pullups.
These days, I do all my pullups on rings because I find it's the most comfortable, and I've also found it has a lot of carryover to regular pullups when I've tested them.