Is the rail available to travel from the nearest airport? How far is the airport from the Beach Park? How can I get from airport to Beach Park for the festival in February?

You can connect to the light rail from the airport and then get off in downtown Tempe and walk maybe a mile or so to the beach park. I don't recall if it stops twice in Downtown Tempe or just once. So worse case about a 2 mile walk. But it will be beautiful that time of year and an easy stroll through campus and downtown.

Very close, you'll have no problems

Yes, yes & yes. Use the metro app to get routes & times

The Light Rail goes from Mesa Arizona to Dunlap