If I leave my headlights on and turn off the car and open the door, the
loud alarm indicator doesn't sound anymore. Does anyone know a fix? Is
it a fuse ?

 

Look for a push switch covered with rubber in the corner of the
driver's seat where door slams. Depending on car year or model, the
position might be a little different. The switch get corroded over time
and stop working. It's pretty easy to open the switch and clean it.