mercur

My friend has a 1987 Toyota 2WD pick up. It has some light problems and I need to know if they are related to each other, or if they are separate problems.

The right turn signal blinks fast, left side blinks normal. The 4 way flashers work fine. The right side tail light does not work at all, but the left side does. Brake lights both work.

RAD4Runner

Your problem is most likely in the BULB or connections or sockets in your right-side light assembly.
Open it up, inspect for broken wires, corrosion, bad connection, rusted sockets, busted bulbs, and correct what you find. There are 2 tail light bulbs in there. One is by itself (the rear side marker light), the other is in same bulb as the stop light (dual-filament).
I after fixing all of the above you still have a problem, break out the multi-meter and we'll go from there.

scope103

Generally flashers are designed to tell you when a bulb isn't working by flashing faster. So if it is your TURN SIGNAL bulb, not the TAIL LIGHT, that is not working, that all makes sense. (When you say the 4 way flashers work fine, do you mean that the left-rear turn signal IS flashing? That would be a real head-scratcher.)

The rear turn signal and brake light share a two filament bulb. If you burn out the turn signal filament, then you have no turn signal (or 4-way) and the turn signal (in the cab and up front) now flasher faster. Just replace the bulb.

Or stand in back and check that the 4-way flashers flash the left turn signal. Switch on the parking lights, and see if the left tail light works.

RAD4Runner

O.P.,
Like Scope says, check each and every light back there.

Chances are the right turn bulb or connection to it is bad, so flasher senses the low current on that side.
When hazards are used there is enough combined current to all 4 corners that the flasher does not see the low current to the right side.