"Has anyone ever made a skinny continuous loop string?"
Lots of them.
"If so do you pad the loops?"
I don't. If needed, I'll just double-serve the loop. A fellow I know did some experimenting. He used clear mono serving (so you could see the padding), a white string, and red padding. In a fairly short time the padding moved to the back of the loop, where it didn't do anything.
"Do continuous loop strings stretch ?"
Short answer, yes. How much depends on several different variables: how it's made (obviously), how many strands, draw weight, temperatures, whether it's pre-stretched or not (and how much tension was used), etc. Endless do settle in faster, but all materials have some degree of elasticity (stretch) and elongation (creep).
I've use original Fast Flight, 450 Premium, 450+, Formula 8125, 8125G, 8190, PENN 66, B-50, B-500, B-55, Nitro Excellerant, 452X,...and I figure several I can't think of at the moment. In 20+ years I've yet to find one that won't stretch at least a little under the right conditions. Of the materials I've used, 450+, 452X, and 8190 have the least.
Chad