Do any of the agencies teach a course in servicing your own dive gear?
PADI offers an Equipment Specialist class that provides an overview of gear maintenance but in no way will train you to fully service your own gear. Done properly, you will leave the class with a good layman's knowledge of how to take care of your equipment.
Can one buy th necessary rebuild kits to service his/her own regulator?
It's easier to do than it used to be but its still a black market transaction - caveat emptor and all that stuff.
Do any manufacturers teach gear servicing courses to the end user?
No. They actively discourage end users from performing anything more than basic cleaning and would rather you leave the service to trained technicians. Servicing your own equipment will void the warranty, in most cases.
What normal type gear needs annual servicing besides regulators and BCDs?
Cylinders and valves need to be opened up, inspected and possibly cleaned. Everything needs a good anuual cleaning. PSI and TDI offer cylinder inspection courses, PSI offers a valve servicing course, TDI and ANDI offer cylinder cleaning courses, for all three you will still need to spend a fair amount of time and money to get started.
Like many things, it's easy to service your own gear badly, harder to do it properly. There are a lot of
expensive and highly specialized tools you will need to have, some experience working with semi-complex mechanical devices and the willingness to devote the time to learning how to do it yourself. For most folks, the rational decision is to have someone else work on their gear.