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No, alarm issue doesn't sound right to me either.
A few things can cause cold start issues.
Fuel injected engines can not use a regular "choke" like a carb could, no jets to suck the extra fuel from.
So they use a "choke mode" in the computer software to set high idle(via IAC valve), rich fuel mix(open injectors longer), and advanced timing(coil control from computer).
Now all this is triggered by the coolant temperature, the temperature is "read" from the ECT(engine coolant temp) sensor hooked to the computer, this is a simple two wire resistance sensor and only used by the computer.
Dashboard temp gauge uses a one wire "sender", they look similar but are not the same nor are they connected together, so 2 wire = computer, 1 wire = dash gauge
On the 4cyl I think the ECT sensor is towards the front of the engine, drivers side just below the head on the block.
But not sure on the 2000
This sensor is easy to test with an OHM meter or volt meter
Look here:
http://oldfuelinjection.com/?p=28
ECT sensors rarely fail, but not never fail.
If the ECT sensor was "reading" engine warm all the time then computer wouldn't know there was a problem, so no CEL(check engine light)
But engine would run poorly until it actually warmed up a bit.
Check a few spark plugs make sure they are the correct ones for your engine, and are in good shape.
If ECT sensor is working then engine should run rich at start up, this could cause fuel fouling if plug spark is not hot enough, so until spark plug heats up you get misfires.
Another issue that is easy to test for is leaking fuel pressure.
Fuel injected engine fuel pumps are not on all the time, the computer turns them on and off according to RPM, throttle and speed.
When key is turned to ON the fuel pump will come on for 2 seconds then shut off.
It will not start again unless engine starts and runs for a bit.
If there was a small pressure leak in the fuel system and truck sits for a few hours, engine now cold, then when you go to start it the pressure would be too low and engine wouldn't start well, the 2 seconds isn't enough to restore full pressure.
So the test:
Turn key to ON, count to 3
Turn key OFF
Repeat 3 times
Then try to start engine
If it starts right up and seems normal then you have a pressure leak, next step would be "where is it leaking?"