Yes, they should be on separate breakers.
Yes. Your dishwasher discharge hose is probably hooked into the side of the disposal. Running your disposal will not hurt the dishwasher.
You can have one or both. If you have a disposal, the dishwasher usually empties into it. This helps keep the disposal clean and any food particles from the dishwasher get chopped up. Any home can have a dishwasher, but if your home is on a septic system, a garbage disposal is distinctly discouraged.
Of course
Yes the outlet for the dishwasher should be on the same side as the disposal,so the disposal can handle the waste water from the dishwasher.
The disposal has to be in place so that the drain on the dishwasher is connected. If you do not have a disposal or are removing it, there is a extension tube for the drain that has the dishwasher tube in it. The disposal does not have to work for the dishwasher to drain. The water is run through the disposal so that any food particles get ground before going in the drain. After a while, the disposal will begin to smell if it is not run.
A fitting made specifically for dishwasher that goes in drain line under kitchen sink.
The sink drain is blocked somewhere after the disposal. take apart and clean or renew sink drain.
Fit a short length of capped hose to the disoposal fixture.
On the side of the disposal, there is a tube, 1/2 inch diameter and 1 1/2 inches long. This is for the discharge line from a dishwasher. The knock out plug in in that tube. If you do not have a dishwasher, you leave the plug in so that the disposal will not leak. Having the dishwasher empty into the disposal helps keep it clean and free of food particles.
Usually, but it depends on your local codes.
If you used disposal tap for dishwasher drain there is a knock-out inside the tap that must be removed.Take a large screw driver and hammer and remove knock-out.
Sure! Should not be a problem. Unless your dishwasher is the type you can put dishes in that still have food on them. I would rinse them first if that is the case.