The LED is not a good indicator for the capacity of the battery. As long as there's enough power to crank the engine, everything is fine. Plus fuel pump will still work even when there's not enough power left to crank the engine.

The LED indicates the status of the immobiliser, just check your manual. And yes, even if the immobiliser still works, the engine usually will crank.

Most obvious reason: your fuel pump gave up. First check the power supply for the pump. If it's fine, you'll need a new pump.