If that center tee has a baffle in it, you're really going to have some messed-up behavior. From what I can see, I'd say get that thing out of there, put a solid arm across, then put the waste into that. I can't speak to the cross-connection problem, but I'm guessing this is done wrong. If you're getting anything out of the spout while the disposer is running, that's just wrong. On the other hand, maybe the water is just shooting up out of the drain.
Is it possible that the drain is clogged and isn't taking all the water, so that the disposer is shooting it out?
The tee at the trap side of the picture must have a baffle in it, and the one in the middle must not - considering that it would be upside down.
In addition, the dishwasher drain hose should run up at least to near the ledge - it's called a "high loop" and keeps water from draining back into the dishwasher from the disposer. Many states require a safe waste on that instead of the high loop.