Electrical Wiring for out-building
No, the 30A breaker is not allowed to protect 12-2 [see NEC 240.6 (D)]. One option is to swap out the 30 amp breaker with a 20 amp breaker. Then you would be okay as it sounds like you only want to run some lights and 110 receptacles. The larger wire is good for 30 amps, but you have to protect the 12AWG with a 20. On the bright side, it is very helpful to have that larger 10-2 wire to cover the voltage drop that would otherwise occur over the distance you mentioned. Another option would be to set a fuse box at the end of the 10-2 cable there in the storage building with a 20 amp fuse to protect your branch circuit. That, however, means the 30 amp breaker and wire converts from being a branch circuit to a feeder and you'll have to drive a ground rod at the outbuilding [NEC 250.32(A) with respect to the exception as well].