Can AWG 6 be used in the middle of a 20a AWG 12 circuit?
There's no problem connecting a bigger conductor, as long as the circuit breaker protects the smallest conductor in the circuit. In fact, that big wire will reduce voltage drop over that 100 foot run. Actually, it's 100 feet, plus the length of whatever wiring you run to the outlet and lights, plus the length of whatever extension cord(s) you plug into the outlet.

If you were to use a 12 AWG wire for a 100 foot 120V single phase run, you'd experience almost 8% voltage drop if you were actually pushing 20 amperes. Assuming you had 120 volts at the source, you'd have about 112 at the other end, which isn't a problem for a lot of loads (aside from using a little more electricity), but for some types of equipment I've seen it cause some weirdness. A 10 AWG wire would experience 5% voltage drop over the same distance, and 8 AWG wire would give you 3% voltage drop. If you ran a 240V split phase circuit down that 6 AWG wire, you'd experience about 5% voltage drop if you were pushing 40 amperes of current.

It seems that this existing wire was probably originally meant to warm up a subpanel in the other structure? Why not just install a small subpanel with a 40A two pole main breaker, then run a couple of 15A or 20A circuits out of the subpanel with a little bit of room to grow?  The panel itself would only cost about $15. The breakers would cost a little more if you're required to install GFCI and/or AFCI breakers.