Can I run two 12/2 wires from one 12/3 wire?
You don't even need to replace the fixture box.  Fixture boxes are allowed to be junction boxes. Boxes are a PitA to change anyway, because of the risk of damaging the cables and the close quarters.  They make blanking plates for unused ceiling boxes.

Also, presuming the box is a fan box, you may want to leave it intact, in case the next owner wants a ceiling fan or a light there.  

A Multi-Wire Branch Circuit is fine as you draw, as long as the /3 cable lands on one breaker whose handles are factory-tied together, or two independent breakers with a field-installed handle-tie that is UL listed for the breakers.  There are a few other requirements for MWBCs, but they are irrelevant to your plan.