Typically the generator gets starting power from the house battery bank. The generator starter motor is often the highest current load these batteries would see by far, at least if you don't have a large inverter; my Onan starter uses somewhere in the vicinity of 100A, roughly speaking, when cranking. A weak battery or a poor connection (positive or ground) will quite easily result in low voltage at the starter. The starter clicking rather than cranking is pretty much proof positive of low voltage at the starter.
The propane detector is probably going off because the system voltage is dipping; many are sensitive to low voltages. Unless it's a propane powered generator, there's no reason why starting it should make any propane flow or not flow.
The generator's gas comes from the main fuel tank (again, assuming it's not a propane model). Typically the fuel pickup for the generator is at a quarter tank level, meaning the generator won't run if you have less than a quarter tank of fuel. This is to prevent you from accidentally running the tank dry with the generator and then not being able to drive to a gas pump.
Sometimes the fuel lines deteriorate with age and get small cracks or pinholes that let air in. Sometimes the fuel pump in the generator fails. Sometimes the carburetors get gummed up if gas sits in them. Those are probably the most common fuel-related problems.
On most of these generators, you can prime the fuel system by holding the control switch in the off position before starting it for several seconds (maybe half a minute). You should hear the fuel pump operate; it changes sound when the gas reaches the carb. The fuel line between the gas tank and the generator can be long, depending on where the generator is located, hence the need for the rather lengthy prime.
It is possible to operate the generator from another battery by connecting it to the generator ground and battery plus terminals (the latter on the starter solenoid). Do be aware that the generator uses battery power to excite the generator field and run its control electronics while it is running and does not have a built-in battery charger. With the house battery, the converter charges the battery when the generator is running; but for some other battery, it will run it down sooner or later if nothing is done to charge it.