It's a Generac repower. Starts then slowly looses fire. What could be causing this to happen?

Check the wiring from the utility through to the autostart control.; make sure all connections are solid and the wire itself is in good condition. New wire is no guarantee, several times I've had wire with a break somewhere in the reel. A non-contact voltage detector may helpful; start at the controller end, go to the other end of the wire, and see where you lose the voltage detect. It will detect voltage up to the break if you can keep the detector close enough to the wire.

low voltage can be from incorrect running rpms
as it is new out of the box I suggest that you get a service under warranty to sort out any problems

Unplug the oil sensor.
The oil sensor is located behind the oil filter. I unplugged the oil sensor and it fired right up. Plugged it back in and it runs fine.
It might be because it is cold.
I don't know why but I turned it off and it starts right up now.

Usually generators that have stood for a very long time loose the magnetism in the ignitor coil magnet. With care, if you can find the low voltage wire into the generator 'motor' you can apply a 9v battery briefly to it and spark it into life. Use the generator if you get it generating, for a good period and don't leave it unused for a long while.
*IF YOU DONT KNOW AND UNDERSTAND ELECTRICITY DO NOT TRY THIS. GENERATORS CAN KILL*

What does the fuse power?

Check the air filter. Then mixture. Then propane level supply. Or reverse the sequence. Which works best for you, go with.

The battery is overcharging either by the external charger or by the engine alternator. This is causing the battery to "boil" realeasing hrdrogen gas. One spark or high current demand from cranking can cause this. Have it checked by a pro

What is the wattage of the hair dryer?. It sounds to that your hair dryer is drawing to much power for the generator.

Check for loose connections on your ignition system, mainly around the coil.

See if there is an oil pressure guage on the dash. When you start the machine it probably has a low oil pressure shutdown which shuts the engine off when you release the start button. This is an interlock. Watch the oil guage and keep the start button engaged until you see oil pressure, release the button and it will continue to run at that point. Your second issue may be a function of the choke. If you are having to choke the engine to start, you may have to play with the choke position to keep from overfueling the unit which may cause it to stall and possibly backfire as it has to much fuel to be burned on the compression stroke, and is igniting occasionally on the intake stroke. Hope this helps. Marvin