There are thin stabilizers, but sorry, I have no clue as to the brands etc. to look for.
Yes, definitely the accufeed foot is what you want.
... from the start, I always used it and the same sort for my Bernina now).
Maybe that was why I got through the Supersize-HSTs without problems.
Tension ... in my mind, I would see no need to adjust the tension.
That said, with the assumption that it is working fine for your regular stitching.