1. no, you don't need horizontal track ends (you just won't be as hip as those with a track frame), but you do need ends that are adjustable. i.e. older road bikes with horizontal drop outs, or you can get a really expensive eccentric hub.
2. i wouldn't recommend flat pedals, you can always ride on the bottoms of track pedals until you get the hang of it. or go clipless. as to the brake …uhh, i ride brakeless.
3. surly makes a thing called the fixxer <http://www.surlybikes.com/parts.html> it converts a shimano cassette hub into a fixed hub. but they also have track hubs.
not familiar with that model of schwinn, but if its an old road bike, it should work. you just need to pay attention to the rear hub spacing to make sure the hub and frame match …or use force to make them match.
…and please no more seal monster...