First of all, you don't pay for checking car seats. So #1 is still a decent possibility.
Second, in my experience of traveling solo with my two kids (which was many), people are very very nice. I got off one plane and a random woman was waiting for me at the end of the jetway with a luggage cart. Security would swoop me into my own line, gate agents would help as much as possible, folding strollers or taking car seats at the last minute. It'll all be OK. And any damage that was done to strollers (I think my giant double stroller got banged up) happened to an item that was already old and beat up by me so it was weak to begin with. Chances are good that your stroller would be fine.
You're going to want a stroller at your destination, absolutely, so no to #2. Is the snap-and-go very bare bones? I had a simple stroller that was a little more sturdy with a shade hood, and I was able to perch a kid up there for walking through the airport. You definitely can't expect her to walk through the airport -- what if you are at the last gate? You could possibly hitch a ride on a handicapped cart, but sometimes they are on a different schedule and you may not be able to count on it. But you could call your airline and find out their policies for those at the airports you'll be in.
You also need to think about who needs to be strapped in on the airplane, how many seats do you have? If 3, then they both have carseats (push the snap and go, pull the toddler in her rolling seat, exercise!) If only 2 seats, is the plan for toddler to sit in her own seat? I think she might HAVE to have a car seat, check with your airline.
Bottom line: don't worry about tough, hurried security or gate check hassles, you don't have to manage those on your own... think about the parts where you are on your own-- your walk through the airport, sitting on the plane comfortably, and getting off the plane smoothly on the other end. Leave yourself enough time to not be stressed and relax, it'll be OK!