No I did not have to be "sponsored." I did not know anyone when I got in. I was accepted the first time but I do know a few guys that had a hard time for some reason. One of them had relatives in the local! Yes there were and usually are more applicants that positions. They accepted 60 of us the year I got in and there were a couple hundred applicants.
Most potential inductees are chosen from applications to the program, and then go through an interview proccess in front of he apprenticeship board. They will consider all sorts of things including your grades, attitude, appearance (you bet it counts!), and your answers to their questions. I doubt they'd take you too seriously if you showed up for your interview wearing bermuda shorts and a flowerdy hawaii shirt.