There should be a alarm by-pass (red button under the dash) located on the drivers side above the brake or clutch (if equipted) peddle. Opening the door (as you probably already found out) will set the alarm off, so it's a good idea to let you neibours know what you are doing so the police are not called on you. Once pressing that button, the alarm and the ignition lockouts will be off and will need to be reset before you can arm it again.
it could be a ground problem from batterie to car body or engine to car body make sure you have good ground at vehicle body
First thing is to identify the alarm system, I.E.- who made it and what model it is? (like Viper, Automate, etc...)
Contact the aftermarket alarm manufacturer or the person that installed the alarm (if you had/know who installed the alarm system). Most major system have a phone number and a list of qualified installers for their systems. They will be able to direct you to a shop that can service the system. If your vehicle won't start, you may have to have it towed to the location to diagnose and correct the problem.
Check to make sure you have power to starter with multimeter at starter. If there is, replace starter. If not, trace ignition circuit back from starter.
This is a factory system? don't remember any LED in a factory system.
Open drivers door and look at the hinge area. You will see a black rubber accordion looking tube going from the door to the a-pillar. This is the protector for the wiring harness. Wiggle it and see if this recreates your concern. If it does then pop it loose at both ends so you can inspect the wiring.Check each one carefully,feel them and tug on them.You will be looking for one that the wire inside the wiring insulation has broken,the two wires to really look at closley are the YE/BK and BK .
It may also possibly be a door ajar switch sticking and for this take some WD40 and shoot some in each latch on the side of the door and open and close the door several times and this usually un-sticks it.