OK, well the no brake light on front brake is usually the plunger switch. If you look under the lever/master, you will see a small switch that is actuated by the lever. It is likely bad. Though, the oil light makes me think someone tried to do their own wiring and plugged things into the wrong connections. These 2 issues may be related.
No running tail light, I would pull the cover off and check the bulb, may as well replace it. I have had bulbs look perfectly fine but not actually be any good.
Brake light should come on when rear pedal is pressed. There is a pull switch that uses a spring hooked to it and the rear pedal.
I would suggest getting the wiring diagram here and pull the head light out and do some tracing. These issues won't be solved by any battery. They are wiring.
The ride IS the adventure. The destination is just to get gas!