Quite a bit happening here.
Your issue(s) may all have a commonality (something with the alarm system )
But, lets look at the first two issues first.
1- When I turn the key to start, the engine cranks, but does not start. After a few attempts, it starts up.
2-When I turn the key to start, the engine starts, but stops immediately. After that, it takes a few tries before it starts (as with the 1st behavior)
This could be due to a fuel delivery issue.
You can test this a several ways but the two easiest:
Cycle the ignition several times to the run (NOT START) position.
Then attempt to start the engine.
If that works consistently, more than likely you have a failing fuel pump or perhaps a faulty pressure regulator.
The other method would be to hook up a fuel pressure gauge on to the in bound fuel rail and see what the pressure is when the ignition in in the run position.
3- When I turn the key to start, nothing happens. All the lights stay on on the dashboard, but nothing happens. Exactly as if I were trying to start without pressing the clutch pedal. I have checked the clutch pedal sensor with a multimeter, and it works fine.
This may well be a intermittent contact in the clutch safety switch.
Is there any other activity (fuel pump priming, relays engaging ) when it fails to crank?
Reason: you may have a faulty ignition switch causing this issue also.
I would remove the safety switch and by pass it with a jumper and see if that cures the issue.
As to the alarm issue…
I have no idea, could be faulty wiring, a bad sensor etc.