It's different depending on the aircraft. Generally, strollers, wheelchairs and so on are loaded directly into the hold from the jetway. I'm not sure if they go in a container or in a special container but they are separate because they're loaded again, back on to the jetway on arrival.

Yes, gate-checked items can be damaged. A lot of us have had strollers get a bit bashed up. There are some ways to avoid this. Bagging the stroller keeps it clean and dry, but it wont really help with damage. In some airports, gate-checked items are exposed to rain, etc.

Foam handles often get torn so obviously covering those is a good idea.

Double secure your stroller with a bungee. Someone posted that they're not allowed on some airlines so bungee than bag, so that no one is hurt. The point is that most strollers have flimsy latches that can come undone easily. A half-opened stroller is more vulnerable to damage. You don't want some time-pressed baggage guy trying to get it folded down again!

For your information, if your child has a seat on the plane and the car seat is in "harness mode" (if s/he still fits it this way) then you can use it on board. Best way to get it there in one piece. Don't worry if it's slightly over the seat width. Raise the armrests.