1. I'd very much recommend getting a cover for the car seat, even "just" gate checking it. The fabric can easily rip on the seat, or 1 drop or a short amount of time in the rain and it'll be an unpleasant way to start your trip.

2. Shouldn't be any problem to check both, from my experience. Actually, a few weeks ago, the night before my daughter and I left on a trip, my wife realized we loaned out the stroller we like to travel with. My wife was flying to meet us ~2 days later. She took the stroller with her (traveling alone) and just went and got a gate tag for it at the departure airport. No questions of "where's the kid", etc. So you and your hubby can just each go get a tag. Worst case scenario, then you have them check one, but shouldn't be an issue.