Key fob possible issue

Alarm siren batteries may be shot and you might need a need a new siren. Is the LED going out?

 

Four chirps upon locking indicates that a door, engine bonnet or trunk bonnet isn’t closed properly, or the switches/sensors on one or more aren’t closing the loop.

 

The roof has nothing to do with the alarm unless you’ve flattened the car’s battery. Is the crank healthy? Can you measure the battery voltage during cranking?

You have a USA 1998 355 F1 Spider, I assume. The F1 TCU controls the starter relay, so the inputs into the F1 TCU seem ok. If I remember correctly, the Immobiliser enables the circuits in the engine Motronic ECU and the ECU sends an OK signal to the F1 TCU. The F1 TCU adds it's own logic into the mix (gearbox in N, brake pedal pressed, etc) and if ok, the starter relay is enabled.
If the engine is cranking, it sounds like the Immobiliser is operating correctly (and at least some parts of the Motronic ECU). With the ignition on, the Motronic ECU should enable the engine sensors and arm the fuel pump relay. During cranking, the ECU enables the fuel pump relay.

The symptoms keep changing, so I don't know if you have one problem or several. When you say it won't go off, do you mean the LED won't go out and the car won't crank or something else?

(EDIT) Sometimes the Immobiliser thinks it's in panic mode and requires a long push of the fob button to get it out of that mode.

 

This seems unlikely because the alarm is letting the car crank. No firing may be an ECU issue, a timing issue, fuel supply/pressure issue, electrical power issue....

If you do a battery reset, then turn the ignition on (not start), do you hear the fuel pump priming for a few seconds? Note that the pump is not all that noisy, so turn off your aircon and radio before doing this. Fuel pump priming does not rely on the immobiliser (I confirmed this on my own 5.2 car).

If no priming, check your fuel pump fuse (in the passenger footwell) with an ohmmeter. Reseat the M and N relays. There was a gentleman with an aircon issue on the 360/430 forum, recently, who spend days trying to diagnose his problem. He said he checked the fuses, but, we discovered later, he only did it visually (and missed the break in the fuse). If in doubt, swap it with another (non-essential) fuse.