How do beets bake up in flour?
I haven't tried baking with beets yet, but, as you said, it must be a bit like carrot cake.

To keep my carrot cake from going soggy while baking (carrots giving off steam), I mix them first with the recipe's sugar to weep them. This also pulls out flavour from the carrot to the cake batter. 

With beets, the procedure could be called bleeding rather than weeping, ha ha.