you might want to reconsider a new tire. if you want to stick with tufos, the s22 is a tire with a thick tread, perfect for commuting. the s3 pro is actually a thin, light, low profile tire typically used on TT bikes. i had s3's on a pair of deep carbon wheels before, and they didn't last very long; i already have a bald spot on my rear tire after a quick emergency brake during a road race.
as for the rubber protruding out the side of your wheel, since you said it's thin and rubbery, it's probably the tube inside the tire. maybe you got unlucky and picked up a defective tire. i'd pull the tire off and visually check it, or have a friend fill it with air and inspect the rubber that's sticking out for leaks. my guess is that the sewing job was done kinda half assed in that spot and the pressure caused the tube to blowout.