No, you do not need a garbage disposal to have a dishwasher. A dishwasher will work without one, and while it's rare to see a home with a dishwasher that doesn't have a disposal, it is possible. The two appliances pair well and are often installed together because a disposal simplifies meal cleanup.

However, your dishwasher should not be used as a replacement for a garbage disposal. Subjecting it to continuous food leftovers can lead to major clogs. If you choose not to install a garbage disposal, you should thoroughly rinse your plates or scrape excess food into the trash or a compost pile before loading them into the dishwasher to prevent plumbing issues.

Once you have your garbage disposal disconnected, you can remove the mounting bracket by unscrewing the bolts that are holding the bracket. You can back them out just enough to pop off the snap ring, which prevents the bottom portion of the bracket from disconnecting from the top when you unscrew the bolts. Use a regular screwdriver to pop off the snap ring, and the rest of the components should fall off by themselves.

There are many maintenance and repair issues that lead to a malfunctioning garbage disposal. Regular clogs place wear and tear on the motors, leading to an eventual shutdown. Electrical issues at the breaker, wall outlet, or disposal itself can fry the unit. The most common cause is simply a worn-down motor. These motors can only handle so much use before powering down for good. 

Your garbage disposal is not draining because it might have a clog due to food waste, a foreign object, grease, oil, fat, or it may have a broken motor. When water doesn't flush out of the disposal, that's a key sign that your disposal has a clog or needs repair.