What's the best way to store a cake that's covered in MMF? Should I just leave it out? I don't want it to go mushy if I cover it, but don't want it to get hard if I leave it out.

I leave mine out in a cool pantry - MMF doesn't get hard.

It won't get soft if it's loosely covered. If you put it in an airtight container then it will get soft. It also won't get too hard sitting out. The outer surface will feel dry but the MMF will stay soft enough to cut.