Codes differ everywhere, - however if I as renovator install these appliances, I always put each one on a dedicated separate breaker.
Yes, they should be on separate breakers.
Usually, but it depends on your local codes.
Yes. Your dishwasher discharge hose is probably hooked into the side of the disposal. Running your disposal will not hurt the dishwasher.
Of course
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.
Yes but the breaker would be at your electrical panel, they do not have a breaker connected to the garbage disposal but they do have a reset button on the bottom of the garbage disposal. Most modern homes also have a GFI outlet linked to the garbage disposal. If your disposal is tripping breakers or GFI outlets I would consider that something is very wrong with your disposal, or you installed one with to much power!
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.
each needs a separate 20 amp feed
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.
Your kitchen sink should already be vented if there is a p-trap draining to the wall. if it is an s-trap it is not vented and not to present code requirements. The dishwasher drain goes into the side of the disposal and the disposal should drain to the trap. A separate vent is not required for the disposal or dishwasher as the trap for the sink is what requires a vent.
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.