Fuses keep burning out; rear lights don't come on when brakes are applied. Is there a connection between these two problems?

There are three fuses involved.
There is a 25 amp fuse before the brake switch located in the fuse box under the hood. Then there are two 15amp fuses after the brake switch in the box inside the cabin. One is for the lower brake lights and the other is for the high mount light and the trailer connector.
You would need to know if the 15amp fuse for the lower lights has power when the brake pedal switch is applied.

Try having the alternator checked.

Sounds like you have a blown fuse. Check owners manual for location of fuses, should be under steering wheel.

feed supply problem somewhere ?but where ? try checking the wiring loom at the rear or as its called in american "pigs tails"

The rear blower and the brake lights might not be related troble. The rear blower has a ballast resistor that can fail and it's next to the blower motor in the rear heater housing. The blower motors are also known to go bad so wrapping on it with it turned on might( wake it up) and it'll start woring again as a poor connection shaken or the worn motors brushes shaken down where they make connection again. Then you say the rest of the lights work do you mean that the brake light bulbs come on with the hazard swich entergized? or do the rear lights just light yellow turn lights only? I'd check or even replace the brake light buld=bs before I'd do anything else.their are other things that don't work if the fuses were involved and the same goes for the heater rear blower as the front blower uses the same fuse.

The power flow for the brake lights goes like this: Fuse ->Brake switch -> Turn signal switch -> Rear lights. The only difference is, if you have a 3rd brake light in your rear window, that gets its power directly from the brake switch. It does not go thru the signal switch first. So, if you are having a brake light problem, and your fuses are good, check the 3rd light in the rear window, if it lights up (assuming the bulbs are good, noone ever changes them), then the problem is most likely in the switch, hope this helps!

Look under the dash, on the brake pedal arm. There are one or two switches that activate brake lights and/or shift unlock. Check for loose mounting nut or a missing rubber nubber. Is the brake pedal coming all the way back up? Look on the floor for something not usually found there. These switches are adjustable and it may be as simple as adjusting them. Have a helper look for brake lights on and shift lever unlocked while you push and release the switches with your finger. Depress the brake pedal to get it out of the way. Remember to set the park brake first. Could just be a bad switch or loose connector. Check fuses and the bulbs. Is shifter fully in Park?

All this stuff runs thru the multi-function stalk. (the turn signal lever) It could be worn out.

Yes. The reason it looks like the lights on the rear of the car and the reason it comes on every time you are hitting your brakes, is because your lamp-out module is trying to tell you that you have a brake lamp that is not working. This is normal function if you ave a burned out bulb or a broken wire in your brake lamps. It does not set any codes, so there is nothing to "reset". Just fix the brake lamp and the warning lamp will go away.

If the center high mount brake light comes on when brake pedal is depressed, and the rear brakes don`t come on, and the fuses are not blown, then I would replace the rear bulbs.

Check your ground connections. If you're burning out bulbs all the time, it's usualy due to a bad ground. Remove the rear fender plasic inside panels and follow the bulb harness to the ground connection. Check for corrosion, if found, clean and use dielectric grease of the ground connection.