Why do the break light and battery light come on at the same time. Then after driving for a while they turn off.

run the fault codes to determine the failed sensor that put the light on
could be egine/transmission mounts or cv joints in the drive lines

Have the check engine light scanned for codes related to this problem first. You may have a charging system issue causing low voltage resulting in the lights flashing/stalling.

Combination of brake and battery light mean the alternator is not charging. after you wait for a wile the battery recover some power and will run again but as soon the power drop below 9V the engine die. Test the charging system and replace the alternator
Good Luck

If you mean the brake light knocks out the tail light, the problem could be a bad ground in the light socket.

Won't turn over or won't start?

Has to turn over fairly fast

Check for fuel pressure & either spark or
injector pulse while cranking

It sounds like the serpentine belt has come off.

Sounds like brake light switch just above brake pedal is bad. If you unplug it, you might as well replace it your self.

disconnect all the connections to the battery and clean them , also clean all the grounds, check the alternator to engine ground, and engine to body grounds, what you describe is indication of a bad connection and applying the brake is enough to to lower the voltage needed to power the ignition circuit, brake light use a lot of juice

Your problem should be that the rubber stops that are mounted at the top of your break pedal came out or are worn down too much to press the buttons all the way in.
What they do is push the button that switches off your break lights when the break pedal is released (yes, kind of backwards way to design it, i know). That's why your lights stay on even when your car is off. I had to go to the junkyard to find replacements for mine and they are a ***** to replace.
They are chipped away easily by the break switches when your car shakes badly over a prolonged period of time. Like driving a couple of weeks on a bad front tire.

check and make sure the drive belt did not break thats what is sounds like.