Well, it won't fit on the breaker. Or a 20A receptacle.
You will need to pigtail the 6 AWG down to a 10 AWG to get it to go on a breaker, and 12 AWG to get it to go on a receptacle.
Since it's all 10-12 AWG, and that's allowed 20A, you might as well use a 20A breaker.
15A receptacles are allowed on 20A breakers. (UL requires they be internally rated for 20A). The only real difference 15 vs 20A recep is the T-shaped neutral.
Note that if the wires are black-white-bare, those must be used for hot-neutral-ground, even if they were't before. That means you may need to move the hot wire off the breaker.
As far as reusing the panel, most 60A panels are way too small. If there's anything we've learned around here helping people, it's that people always run out of breaker spaces much sooner than they expected. As such, a subpanel is NO place to scrimp! Buy a huge subpanel that will far exceed your needs, that you'll never, ever fill up. The cost is small - spaces are cheap.
As far as the subpanel in the garage, honestly, I'd consider leaving it put. The X-factor is electric cars. They're really taking off, for real this time! :) As such, having 50A+ in the garage adds value to your house.
I'm sure the 60A garage panel is a tiny thing... I'd have you spend $60 or so on a new 20-24 space panel. I'm very reluctant to "spend people's money" unnecessarily, but this is one cost you will not regret in the long term.
As far as a 15A circuit in the garage, just add a 15A breaker to the garage subpanel.