Alarm problem....

I think the fact that nobody has replied is telling you something.... very strange problem!
This looks like a case for a professional, IMO.

My theory though is that you have a faulty pin switch on the door or one of the lids (or could be the anti-tow sensor). They are falsely sending a trip signal to the alarm and it's setting off the unit. When you set your alarm it should chirp only once, any more than this and it's telling you something is already tripped, IRRC.

Very odd that the fob does nothing when the alarm is sounding.

Again, I'd take it to a professional.

 

OOOOHHHHH!!!!!
Ok, now you may be onto something. I recently replaced my alarm unit (pretty easy project). In my case my battery was utterly flat and my alarm produced no sound at all.
Perhaps your battery is in a weakened state and the alarm unit is on it's way out, thus it's behaving in a faulty mannor? Just a hunch.

And, keep in mind that the small battery inside the unit is rechargable, so if your alarm had been sounding for a while without driving the car (driving the car should recharge this small internal battery from what I understand), then perhaps it's run down?

I've also gathered that these alarm units only have a service life of a few years.