Yes. Your dishwasher discharge hose is probably hooked into the side of the disposal. Running your disposal will not hurt 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.
Yes if you do not you could break your garbage disposal.
Check and see if the water line going from the dishwasher to the disposal has enough slack to run up and over the connection to the disposal.If its just running along the cabinet floor and straight up into the disposal that's the problem.Its called an anti siphon loop
Your air gap is not supposed to leak. Check the hose that leads from the air gap to the garbage disposal for a clog. This should resolve your problem.
Both
run cold water thru it for a minuint after you are done with it, cold water should be running while you use it also,
Depending on how much stuff you would typically run through your disposal would be whatever power garbage disposal you need to get.
Put ice cubes in disposal and run water as usual until ice cubes are shredded.
To deodorize a garbage disposal you can either put some baking soda in the drain and let it sit for a while or my favorite is to run lemon peels through the disposal after you do the dishes every night. You can use orange peels, too. Just make sure the pieces are small enough to run through without clogging it.
If it's truly horrible, dilute some Clorox or other bleach with water and pour it into the non-running disposal. Let it set for about a half hour. Add some ice cubes and run the disposal with cold water. Afterward, you can run ice cubes with rinds from oranges or limes ... any citrus will do ... with cold water. Washing food down the disposal with cold water instead of hot will prevent future disposal stank.
I strongly advise against using drano. The drain line from the dishwasher to the garbage disposal is pretty short. Try running your dishwasher and fill up your sink, when your dishwasher starts to drain, pull the plug on the sink and start your garbage disposal. You may be able to get away with using Drano a few times, but the thin plastic tube will be made brittle and possibly break in the future, also you will want to run it empty a few times if you do use a chemical. Odds are the clog is at the drain and you have some clearly nasty junk clogging it that will require some manual work. I recommend cutting up a coat hanger and bending a small hook on the end to clean it out. There could be other electrical / mechanical issues with your dishwasher, you may want to have a trusted service technician take a look.
You can run it straight through, but the whole reason for running it through the disposal is to catch anything comming off your dishes before it can clog up the pipes.