Can I use a 20 amp and a 15 amp breaker on one 12-3 wire split into 2 circuits with a common ground
I believe that it would be against the NEC to have these two seperate circuits with a shared neutral and not have them on a double pole breaker (or at the very least a handle tie between a 15A and a 20A breaker).

Branch Circuit, Multiwire. A branch circuit that consists of two or more ungrounded conductors that have a voltage between them, and a grounded conductor that has equal voltage between it and each ungrounded conductor of the circuit and that is connected to the neutral or grounded conductor of the system
210.4 Multiwire Branch Circuits.
(B) Disconnecting Means. Each multiwire branch circuit shall be provided with a means that will simultaneously disconnect all ungrounded conductors at the point where the branch circuit originates.

A double pole breaker will simultaneously disconnect both circuits and satisfy the code requirement.
20A breaker is required and 12 gauge copper 10 gauge aluminum is the minimum depending on how long the circuit run is.
Furthermore even if it is on a 20A circuit, you are not required to upgrade your 15A outlets to 20A outlets.  This is only required if there exists a single receptacle on the circuit.  One can only plug in 15A max devices into a 15A outlet, therefore the 20A load circuit load would not be achieved on a single device.
It might also be a code requirement now for a detached building to have its own grounded sub-panel as well but that I will have to look into.