Typically it's going to be an adult female and current litter if we're talking grey squirrels. She'll keep any others run off. If a large enough structure they might subdivide it with different females occupying different locations but this is rare. She may be anywhere along in the process from prepartum to post. You won't know until you have her in hand. If post partum but still early you may be going into the attic to retrieve pinkies. If later they may or may not come out on their on, there is a short window where the young are mobile but will go no further than just looking out the entry. Past that point they will come out to get caught as well.
When you catch a big male, usually after a few days of no activity, on his way into the structure you know that the job is done.
Fox, and reds I have no experience with. We got fox but they behave themselves here and give people and structures a very wide berth, you won't even see one in the yard.
Fliers will vary by season and all noise will be noturnal. Summer breeding, a pair and very rare for them to enter during sommer. Winter, the entire flying squirrel clan will take up residence huddling for warmth so its trap until none. Traps should be victor rat snaps inside on known travel routes, the tight spots like eaves, and on latrines. You can ring a hole with snaps or use a small coni like a 55. Sealing the structure is the only way to stop them, one way devices work.
Not sure if it made it to the archives but Jonesie, Ron Jones, did up a really good grey squirrel thread in the ADC forum called the six stages of the grey squirrel, or it may have been a podcast on the old Robb Russell site.