I'm not a hand quilter, so I can't really speak to the frame leaving marks in the linen. My instinct, however, is that you should be able to steam the quilt after finishing it (since you don't want to wash it) and that should help with wrinkles, etc.
As to the tailor's chalk - there are 2 kinds, one for wool (for marking men's wool suits, etc) and one for cotton, which brushes off the cotton. They're not really interchangeable. The wool kind contains wax, which, when heated with an iron, absorbs into the wool and disappears. If you use it on cotton and hit it with an iron, it will melt and most of the time leave a permanent mark. I'm not sure which one would be best for linen. (Is your linen 100%, or is it a linen-cotton blend?) I think your best bet is to test whatever you want to use on scraps of the linen and see how it responds to brushing off and ironing.