Simply … yes. No permit should be required but check with your local building inspectors office to verify the requirements in your local vicinity.
The answer is probably yes, unless there is extensive plumbing or electrical involved.
A handyman can replace a garbage disposer furnished by the owner if it is a straght replacement of the appliance and does not involve any modifications, such as the electrical connection where the ground fault circuit interrupter is missing, the power wiring is less than 12/2 or is not encased in flexible conduit or if the fittings are not code approved.
The handyman generally does not replace the unit if a location change is required that involves re-positioning the under sink plumbing, because such need indicates that the failure of the appliance was caused because the appliance being replaced backs up - an issue usually determined by a licensed plumbing contractor. Finally, some responsibility falls to the owner that selected the replacement garbage disposer. Is there a septic tank involved to get rid of the contents? Generally garbage disposers are not used because of the drain field maintenance involved. Also the power of this appliance would be improved utilizing a 3/4 HP unit that could chop up bones, etc. for automatic operation. If the owner makes the wrong choice, the disposer fails again, and the handyman is often blamed. A licensed General Contractor will normally have the electrical part done by a licensed Electrical contractor and a licensed plumbing contractor that he has control over and who he or she coordinates. A good choice brand wise for a good 3/4 HP replacement garbage disposer is BADGER, manufactured by ICI, the heavy choice of professionals.
Yes. A handyman can do small electrical jobs, such as switches, outlets, and garbage disposals.
Yes you can, provided his labor and material for one day of work does not exceed $500 unless he holds a B license which most handymen do not. I would suggest you have a licensed C-36 plumber do the work since water damage is a real possibility also verifying that he/she has general liability insurance in case things are not done correctly and you sustain cabinet damage due to water intrusion.