I breastfed, but know a lot of mums who had to start chemo on day 1 after birth, and many others on psychiatric drugs that passed through milk, a few with bone disorders/calcium issues that were instructed not to BF under any circumstances, and many women who had undergone surgery (reduction, mastectomy, implants) that made nursing impossible. (Yay the joys of working in a hospital). Add that into the military mamas who were deployed right after giving birth and the number REALLY climbs. (also the joys of being a woman in the military... there's no pumping and FedEx'ing from a combat zone).
Anyhow....
The short answer to all 3 of your q's is "yep".
1) Milk coming in ... Probably. Colostrum - absolutely, Milk - probably. Binding and cabbage leaves help.
2) Engorgement... if your milk comes in... yep. Binding and cabbage leaves help, but binding also increases the risk of mastitis.
3) Mastitis... it's actually possible to get mastitis at ANY time (aka neither nursing nor pregnancy is required), but it's unlikely. Your biggest risk period for mastitis will be in the first month when your milk attempts to come in and the ducts get clogged.
Some women I know go immediately on high dose birth control pills to help stop their milk from coming in... but it's iffy.
And for pushy nurses... here's a phrase:
"Pop quiz... How many medical conditions do you know of that preclude nursing?"