2000 Mitsubishi Galant Brake light/Battery light on togethe

when 1st start the engine, this two: (Brake light/Battery light stay on alone together. But after engine running smoothly down, this two lights disappear and coming back on everytime you step on the gas pedal and also cause the more you speed, the more my temperature gauge to jump higher near the (Red area). But coming back in the middle when I stop.

Note: No fault codes stored!