I'd disconnect the switch at the clutch lever, the one on the neutral switch and investigate the fuel sending unit. Figure out why one isn't working, fix it and move on to the next. There's only a few on the bike but one is causing havoc with your gauge pod, tach and your ignition system for reasons I can't imagine but it seems you have a bag of worms and the only way to get it sorted is one at a time. Some where in there you'll find the culprit. Maybe look for wire tampering that's not factory and investigate.
Someone probably put the bulb plug in the wrong hole on the pod, with a new set of bulbs in all sockets function check everything. Fuel and oil should come on momentarily on start, green neutral on when in neutral. If the fuel light comes on than the sending unit is working, Probably won't since you stated it never did. A new sending unit is in order and easy to replace, cleaning the switch is also simple as someone else stated on here, it may not be making good contact.
The bike stalls when put into gear tells me you have a safety switch out of adjustment or bad, you can jump them to trick the engine to run. one at the clutch maybe? Or the one on the side stand.
The switch (res/on) does not control the fuel light, it only returns power to the pump giving you the last half gallon of fuel. If the amount of fuel is above the top of the sending unit in the tank, it will go out, if no, it will come on. look into the tank to see.