All three brake lights don't work

sounds like you may have a electrical connector that is not making a good connection, possible causes are loose connector-wiring is broken-wiring is over heating and the plastic is burned and /or the wiring pins are oxidized and dirty from heat .

I can almost 100% guaranty the turn signal switch is the problem. Power for the brake lights starts at the brake switch. power then splits from there and the third brake light gets power directly from the brake switch. Since the turn signal and the brake lights are on the same bulb power for the brake lights is routed through the turn signal switch. the switch depending on what position it is in will allow brake power or turn signal power to there respective left or right brake/turn bulbs.

If you hold your foot on the brake pedal and wiggle the turn signal and or play with the hazard switch the brake lights will almost always intermittently start to work. this will confirm a bad turn signal switch which is pretty common on GM vehicle with this style setup.

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The anti theft system failing and disabling the starter.... Just a thought

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See if bulbs have burnt out in the left brake light and rear turn signal (one bulb if combined) and also rear center high mount brake light.

If the left turn signal has a slower speed than the right turn signal, this is a symptom of a bulb or bulbs burnt out.

Usually, the wiring is not a problem, but I would not rule wiring (or sockets) as a problem out.

Good luck on this repair.

Check the brake light switch on the top of brake pedal, it's most likely the cause.

The bulbs then to burn out when cars reach this age. The bulbs can be replaced, most cars use 194/168 and 74 bulbs in the dash.

this is very common problem, i ve repaired so many, you will need to replace the turn signal switch, but make sure the brake light bulbs are good first, if they are good then its the switch, to test this step on the brake and gentle wiggle the signal lever up and down and back and fourth and watch the brake lights lite up. hope this helps.

its caused by a switch in the steering column. I have the same problem in my 1995 Lumina. I get around it by just pulling back on the turn signal lever when I hit the brakes.
Other wise you have to replace the switch. there are solutions posted on the internet.

this would be a multifunction switch which turn signal switch wipers ect. run through located on steering column under plastic cover.assuming the brake light switch is good see if third brake light works when you press the pedal if so it is deffintly the multi switch.

This is a classic example of turn signal switch failure. On GM vehicles with a multi-function turn signal switch and combination stop/turn/tail lights (all rear lights are behind red colored lenses except for back-up lights) The brake light power is routed through the turn signal switch. This makes it possible for the right rear light to function as a stop light while the left rear turn signal flashes, and vice-versa. When the contacts fail to remain constant the brake lights, and turn signals can become intermittent. The proof that the brake light switch is functioning correctly is that the high center (cyclops) brake light functions properly. Wiring integrity is verified if you can prove that turn signals operate, but that can still be intermittent with this complaint, BUT if you use your hazard switch to turn on all the signals at once, you can verify wiring integrity. This is because the turn signal, and the brake light use the same circuit, and bulb filament in the rear lighting assembly. If any Hazard lights fail to function, check bulbs first, then retest. If all Hazrd light flash correctly and reliably, bulbs, sockets, and wiring have proven integrity, and the turn signal switch is the remaining component in the system which needs to be tested. Unfortunately, this switch requires disassembly of the steering column for service, testing or replacement. If you have never disassembled, and successfully reassembled a GM tilt steering column, I would reccommend professional assistance.