Your ambient air temp sensor DOES affect your COLD START enrichment map program running inside the Motronic ECU (Engine Control Unit). If its reading -37 degrees that means it is either missing or has a broken/disconnected wire or in other words infinite resistance. Fix it you will have fewer problems, they're cheap.
The Trunk issue is not electrical if it fires all the door locks when you rotate the key in it. Are you pushing the button in while the key is turned? My 87 if freaky that way while my 86 is not. DO NOT spray WD in the lock mechanism unless you want all the electricals to fail.
Sensors on these cars, and ANY vehicle with a computer, are VERY important since that is how EVERYTHING from your fuel maps, dwell timming and on and on and on are controlled. Without the proper information from ALL of the sensors the computer will simply hunt and peck to get it to kind of work if at all. Hence the long cranking sputtering and crappy start up quality. In fact when starting one of these cars its best to turn the key on hit the brake, push the check button on the overhead and then crank, giving the computer a few seconds to get the sensor data and set the start up map before you try to start it. Your car will actually start faster and run better if you do that.
As far as the crank sensor is concerned, if while cranking the engine, the check engine light stays on, then the vehicle is not getting a signal from it and will not start at all. If it loses the signal while driving it will stop running. If the car starts and runs and doesn't simply sputter out intermittently then there is nothing wrong with that sensor/wiring.
Other critical sensors are you mass air flow wich includes your baro sensor, and your throttle posistion sensor. They are both mechanical sensors and do wear out. Also make sure there are no leaks in the vacuum system or the rubber elbow boot between the maf and the throttle plate, it should be tight and fully intact. AutohauseAZ for like 27 bucks or something for it for a new one.
Make sure ALL the sensors read and the hookups are secure and tight in the vac and intake system, most of your problems should disappear. As far as water getting into a cat and freezing especially a quart!? that sounds like a head gasket issue as any water in front of the converter would HAVE to go through the engine, and that you would notice. Good Luck