It can only be either the distributor or the over boost protection facility, or a faulty ECU. If it does it randomly then it could be wiring.

The fuel pump relay just cuts the fuel not the ignition (which is why you lose the rev counter).

The battery light can be either the blue wire which runs from the back of the dash clocks to the alternator, the bulb in the dash or the alternator itself. The car will not charge without the battery warning light so you may break down some day if you use it without the light working.