I have a 686 with about a .010" gap between the frame and barrel/lug shoulder. Wondering how much one turn would be.

The barrel tenon is 36 tpi. One turn equals 1/36 inch, or 0.0277" or 27.7 one-thousandths of an inch. If you are thinking of setting your barrel back one full turn and then re-fitting the barrel-cylinder gap to 0.005", that is how it is done.

This one is basically NIB, only fired once. I wonder if S&W would fix it for free?

If it's .01, they will say it's within tolerances. If an older gun, they will charge around 100 bucks.
If it spits or bugs you send it in.