This sounds like a serious wiring issue. The fuel tank level sender shouldn't have anything to do with the engine electrics, expect that the key ON position provides power to the gauge.
The wiring for the fuel pump is 'double source'. The first source is a post on the starter motor labeled 'I'. It will have a purple/yellow wire. This provides the pump power while cranking the engine (and before the oil pressure comes up). The second source is the oil pressure switch. Once 4 psi of oil pressure is applied the switch closes and powers the pump. Just be aware that once that switch closes, power is then on the wire that goes to the starter motor. And the converse applies, when the switch closes, the rest of the system connected back through the switch is then powered. Unfortunately I saw a tech using a remote starter switch and leave the key OFF to crank the engine, thinking by leaving the key OFF would stop sparks. When the oil pressure switch closed, he got a very big shock!
Chris........