1994 Chevy S-10 Rear brake lights and turn signals do not w

Electrical problem
1994 Chevy S-10 6 cyl Two Wheel Drive Automatic

First; The right side only was out, both brake light and turn signal. Green indicating arrow (instrument panel) on the right was solid green, left side flashed.

Next; I replaced the bulb and the relay (flasher). Then NONE of the rear brake or turn signals lights work but both green indicating arrows are flashing? I checked the ground and traced most of the wires and they look good. Brake light switch seems fine (I hear a click when I move the switch without depressing brake pedal). Strange, rear lights come on with headlights but no brake or turn signal.

Now; My car now will not start, seems like the battery? Would the battery cause the light problem? If battery has nothing to do with lights, what else could be wrong?
Can you please help me?