Subpanel box wiring

Your 10/2 can be properly used to provide 30 amps @ 120V. The 10/3 would provide 30 amps @ 240V. So the question is what will the peak load be for the 3 circuits you want in your outbuilding? If it doesn't exceed 30 amps, then you'll be fine with your 10/2.
If it does exceed 30 amps, you'll need to pull a different wire, e.g. 10/3. With a 240V feeder, you can divide the 3 circuits among the two phases, so that the peek draw on each phase never exceeds 30 amps. This will require a double pole breaker for the feeder.

While it is unusual, there is nothing wrong with using only one half of a breaker box with a 120V feed. Or with jumping the two main lugs so they are on the same phase. Note that if you are only using one half of the breaker box, you need to look at the bus bar pattern to see which slots to use. It is probably a zig-zag pattern, every other slot in each row, with the two sets offset one.
Cheers, Wayne
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