I'll try to answer your questions as stated.
Minimum height difference
I'm not sure if there is a code for this, but it's good practice to have the bottom of the disposal drain higher than the top of the drain into the wall. So a height difference of at least one pipe-width.
But, as long as the bottom of your tailpiece is higher than the wall drain, the disposal shouldn't hold water. If the wall drain was higher, then the disposal would always have a pool of smelly water sitting in it. Your problem is that the height difference is so slight that the inside of a fitting or even a very small buildup of gunk will cause water to start pooling in the disposal. I would strive to have at least a 3/4" difference.
The depth of your new sink could be the issue. You can look for a disposal that has a shallower neck, or cut into the wall to see about lowering the drain opening there.
Deep P Trap
I worry about the depth of the trap because that is a lot of room for gunk to accumulate. The disposal acts like a water pump when it's on, so it should be able to blast water through that deep trap, but it's not on all the time. Oil, grease, and other yuck can slip down into the trap and congeal causing smells and clogs.
I don't see any reason the trap had to be that deep and I think that can be easily improved on.
I'm not going to fully explain it all here, but that part is fine. The dishwasher drain should either have a "high loop" or an "air gap" (search terms to learn more) to help prevent siphoning of dirty water back on to your clean dishes in the event of a clog, but the height of the drain line entry doesn't matter because the dishwasher does have a pump for the drain. It's pool of water is near the floor, so it has to have a pump to get rid of the water - gravity is not an option.