Can't answer your last one-
1) How large should a run in shelter be for 3 horses? (It is more likely that I will only have 2 horses -- mine and a retiree companion, but I would prefer to build with just enough flexibility to be able to shelter one extra if needed.)
Like the first poster said, 10 (or 12') by 30 feet.
2) How large should my dry lot/sacrifice area be for 3 horses? Am I likely to need to put down gravel or something to prevent it from becoming a mud pit?
Yes you will need gravel. I like pea gravel, its small enough it won't hurt horses hooves, keeps them out of mud, and toughens hooves. I have a 100 foot long dry lot, half 40 feet and half 70 feet wide- Candy will bully Casey sometimes, but no more than just walking around her with her ears pinned. I would recommend 120 by 50 feet wide for 3 horses, as long as they get along with each other.
3) What should the flooring of my shelter consist of? I have considered putting down gravel and perhaps some sand on top... or would gravel with rubber mats on top be better?
I would put down some road base or large crushed gravel, make sure it is packed well, and put down mats. Easiest to keep clean, horses are more likely to poo and pee outside rather than stinking up the shelter.
“Good things come to those who wait… greater things come to those who get off their ass and do anything to make it happen.” - Unknown