[dlgn]

Those are pretty. Is there any efficient general way to solve them? ... or even any general way to denote them? I imagine the designs could get very complicated.

The first 2 I think I could find a nice algorithm for and possibly the next 2. For the rest, the best I could do is randomly manipulate it until it's solved. Bogosolve. Anything better than that requires more thought.

Also, are they functional or do they just sit there and look pretty?

It should be possible to form a group on the set of possible rotations over the concatenation of rotations. So sayeth Wikipedia. I don't know enough group theory to be certain if there's a general way to solve them, though, just to know that they must be solvable. Which is, you know, pretty obvious.