1998 Mitsubishi Eclipse cold start issue

All cars except Chryslers use a mass air flow sensor, and that one has by far, the biggest say in how much fuel enters the engine. You'll experience stalling even if there's a small leak in the fresh air tube between that sensor and the throttle body. Any air that sneaks in without going through that sensor won't get any fuel to go with it.

An intermittent single, rather loud clunk from the starter, once each time you turn the ignition switch to "crank" is commonly caused by worn starter solenoid contacts. That will get progressively worse over the next weeks and months. A bad battery cable connection can cause that too, but that usually won't be intermittent.

Very possible. There's two different types. The most common one has a tiny temperature-sensing element that is heated with an electrical current, then it is cooled by the air flowing over it. More air equals more cooling. The sensor changes resistance with changes in its temperature. That's how the assembly works, but that means incoming air temperature is a variable that has to be taken into account. For that reason, there's a separate air temperature sensor built into the mass air flow sensor. If it incorrectly reports the air temperature is higher than it really is, the Engine Computer won't command enough fuel.

We used chokes on older engines with carburetors because liquid gasoline doesn't burn. It has to be a vapor. In cold weather the fuel vaporizes more slowly. The choke was needed to dump in lots of fuel in hopes enough would vaporize to make the engine run right. With fuel injected engines that's less of a problem but there is still the need for more fuel for starting a cold engine. If your mass air flow sensor is reporting a wrong value, you may get the engine started with a small squirt of starting fluid through a vacuum port or in the air filter box.