Number 1 - if it is on auto. it will take a second or two to come on. If not - mine does the same thing. I am not sure why. Several other friends that have the 126 chassis cars do the same thing.

Number 2 - my parent's '88 300SE has the same problem. It is in the switch in one of the doors - it is connected to the door handles. They elected to just disconnect the alarm - under passenger side floor mat.

Number 3 - could be several things. Have you checked previous maintenance records to see if the plugs, wires, rotor button, etc. have been changed? It could be the OVP (as mentioned in several other posts). It could be a vacuum leak. It might be worth taking it to a reputable independent to see what he can find for you.