I've never had the airline ask but immigration will ask how long and why you're there. When doing the e-visa it will ask your ticket information for arriving and departure. (At least for usa).I had to upload photos of it all during the application.
look for local visa wherever you're staying in thailand. Any day now Thailand will start giving the free 60 days with the new rules and then once you're about a month in you will need to find a local visa shop and pay for the education visa. Price seems to vary by shop but it's typically 25,000 to 30,000 for the education visa and then every 90 days you have to pay about 5000.
It really depends on when you go and for how long. I've always stuck with Expedia to find the best flights. This time it turned out to be United and ANA.
There's absolutely no problem in doing that. The usa passport basically gives you a 30 day visa. Once you leave Thailand the count starts over. I've done it several times between Thailand and manila.