Disposal not working

NO. You CANNOT power a disposal from an existing kitchen counter receptacle circuit.

If the dishwasher is on a 20A circuit typically you can share that circuit, other than that usually they are on a dedicated circuit.

The common disposal unit drain pipe connection looks like the following picture:

There may be some variation in the design of the tail pipe assemblies from manufacturer to manufacturer but the ones for the above type connection generally follow the form shown here:

These tail pipe pieces are generally supplied with the disposer. It is possible to purchase manufacturer specific tail pipes at appliance parts outlet stores and from online sites. It may also be possible to purchase a generic replacement at some hardware and big box home stores. If you purchase a new one make sure to include a new rubber gasket.

It has been my experience that a common cause of leaks at the disposer outlet is the corrosion of the body housing of the unit around the drain hole. In the cases I have dealt with it was time to replace the disposer unit with a new one. One thing to check is if the lower part of the disposer housing is wet or showing signs of rust and corrosion separate from the drain pipe attachment. If so this is often an indication of the sealed bearing inside the disposer starting to leak and is another reason for replacement. On disposers that I have replaced I have noticed leaks from both the bearing and the drain pipe area.