When i turn my headlights on, the back lights turn off.

Check the ground cable from lights to chassis.

Normally this condition is the result of a poor body ground connection. Testing for this can be deceiving. At the static(nothing on) condition, testing can show a good ground. However when a circuit is excited, it has a determined resistance and a set current flow. If the ground connection to this or a branch circuit is loose, corroded, or broken wire. This will allow just enough connection to show as being good, but not good enough to complete the circuit with the required amount of connection when energized.

headlight switch is bad, change the headlight switch itsellf will short out, therefore blowing your fuse. turn signals and break lights are on a different relay, therefore they will still work

I think you have a bad ground at the back on both rear lights or their earth out on the same connection but these american cars use a funny system where the stop lights are the indicators and not a seperate yellow light like europeon motors ,Now these brake light switches also allow the indicators to work by shutting off the feed to the stop light so it flashes with indicators ,Why american cars use this system i dont know as its so complicated for the stop light switch function but try the ground or earth wire on the back lights and if no joy then ask a american mechanic with experiance of this sort of circuitry as when i get american cars for import i have to fit new back lights and disconnect the brake light switch .

I hope I'm wrong, but sounds like a wiring harness malfunction possibility. Back in the late 70's, I had a '67 VW van. At night, when I'd turn off the ignition the engine would turn off, as it should IF I TURNED OFF THE LIGHTS, BUT IF i TURNED THE LIGHTS BACK ON AS THE ENGINE WAS SHUTTING DOWN, THE ENGINE WOULD RESUME RUNNING.

check power wire which runs to the brakelight with a test light, get someone to apply the brake on and off and see if there is power getting back to that point, if no power check the switch on the brake pedal to see if power is getting to that point.
also check the earth on the bulbs because they wont work if there is no earth

Your cruise will not work if the brake lights are on just like the cruise disengages when you hit the brakes.As far as the brake lights with the headlights,sounds like there is something crossed.If you hold your breaklights on with your headlights off is there a glow of the headlights or your running lights.If your rear bulbs are the round push in and turn type make sure they are installes correctly.You may have one where the contact on the bottom of the bulb is causing a jump of power from your running/parking light over to the brake light side.Well hope this helps and good luck.

give the emergency brake a good slap and make sure it's released fully if you put the e brake down a click or two, it will turn off the automatic headlight on day feature . good luck

First thing i would do is chg. the break light switch, and make sure to chk. all grounds around tailights, sometimes if there is a ground problem it will travel back thru the hot wire to get a ground, witch in turn makes everything go wackey.