The towels to rent (and yes I can confirm they cost $1) are much smaller than a beach towel. If I had to guesstimate the dimensions of the rental towel, I'd say they're somewhere near 18"x36". Not sure of the exact measurements, but they are a LOT smaller than the Blizzard Beach towel I bought a few weeks back (to stay dry from the rain! I was on my way out anyway).
As for bringing them with you to the water park from the hotel, I'm not sure if that's allowed or not. BUT, they are the same design, just plain white with ridges on them. I figure they just wash them all together in one location and bring clean towels to the hotel and the water park without tracking which towel "belongs" where.