DP tiling a 2xN tile with L shaped tiles and 2x1 tiles?
Q1. Of course you can consider a third variable in the recursion, but that would not simplify the analysis. In fact from the case where you put the single horizontal domino, the next step would automatically put the second horizontal one below it. So the given recursion takes a little shortcut.

Q2. That is precisely the point. We have to make sure the analysis steps from existing state to existing state, in a kind of unique way. If we overlook a state we miss possibilities. But here we easily see how the states are related, and how we step from one to another.