I've gatechecked a double stroller, (GT2) as has my sister. Also done City Select and City Mini, which are not small strollers.

The carseat gets checked in with the rest of the baggage.

I was never worried about damaging a seat. The airline is responsible for damage, we've had one carseat replaced and my sister had a stroller replaced. For an infant bucket carrier, they're in there for less than a year anyway and pretty hard to damage. We also use a cheaper seat for travel that is the spare one for my pickup truck, it was only $100, and it's super light. AC will wrap the carseat in plastic for some protection, we didn't buy any sort of covers.

If the stroller is gate checked, then you'll wait for it on the jetway. The carseat will at the special/oversize baggage area.