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it sounds like your door switch is defective or needs adjusting. Some cars have the courtesy light switch inside the main door lock/latch mechanism. In this case, often the whole assembly needs to be replaced. You can pull your courtesy/dome light fuse, so your battery won't go dead until you get the problem resolved. There should be a fuse guide in your owners manual and a location for the interior fuse panel. There may be a guide on the fuse panel cover as well. it is usually in the left to lower left dash area.
it sounds like you need to check all your work. ( this happens to Pro techs once in awhile as well. it has happened to me with 22+ years of exp ) Check all the spark plugs and how tight they are ( 15 ft lbs of torque ia a normal value. ) Then check your new cables and see that they are properly attached and in the correct order. Next make sure that your air boot from your air filter to you throttle body is on right and that any hoses that should be attached are properly done so. If these basic checks don't work, then try you PCV valve. Make sure that is installed correctly and that the hoses are not broken or cracked. Maybe even try your old PCV valve, sometimes a parts store makes a mistake and gives out the wrong valve which can make an engine run very poorly.
it is best to replace the whole starter on a vehicle that is this old because the starter motor and brushes are worn as well. ( you will have to do the same repair twice ) You replace the whole starter motor as a unit with the solenoid. You can get a Chiltons manual at an auto parts store to help you. Be SURE and disconnect both negative ( 1st) and positive battery cables before you do anything and be sure that the starter matches up perfectly. there are a few different starter configurations for your Celica.
what does your temp gauge say when the car is heating normally? It sounds like there is a circulation problem that is helped by the faster speed of the engine when you are on the highway. It could be that the impeller of your water pump is worn and it takes a higher engine rpm to circulate the water, especially in a cold environment like winter in ND. Another possibility is that your heater core could be restricted and the higher engine rpm pushes/forces more coolant flow. It is pretty simple to flush out a heater core while checking a water pump impeller is more involved and requires timing belt component removal. I would begin with flow testing the heater core by disconnecting the heater hoses and either blowing through it or using a hose to flow water through it. If it is restricted, then you can flush it with some radiator flush. It is not an uncommon problem, especially if the coolant changes have been infrequent ( like more than every 30,000 miles ).
Believe it or not, jump starting a modern vehicle with several computers can cause a lot of problems. it can corrupt the software in 1 or more of the major computers. I know this because I was a lead diagnostic tech at major German and European Dealerships and Independent Shops. If you don't feel confident about the work being performed at the dealership, go to a MB owners forum online and find out ask who the best MB shop is in your area and who specializes in sophisticated electrical and computer trouble shooting. Excellent independent shops have the where with all to reprogram the computers with the HHT ( the MB diagnostic and reprogramming capable laptop ) Typically if a car has been jump started several times, ( many tow truck drivers are very behind the times in this area ) the computers need to be reprogrammed which can take many hours but should not require too many parts other than an over load relay or 2.
there are 2 fans to aid in cooling the engine during heavy loads and at high temperatures. When the A/C is on ( or the defroster which also uses the A/C system to dehumidify the windshield ) the engine is hit with a 20 horsepower load and the A/C condenser near the radiator, gives off a lot of heat, 200 + degrees on a hot days. During these periods of operation at least 1 fan should run. During the cold weather, it is not uncommon for neither cooling fan to operate because as you drive the wind through the radiator cools the engine and coolant sufficiently. Is your car overheating at all? If you are very concerned about the operation of both the cooling fans, a good mechanic could use a scan tool to verify that both fans do in fact operate. My 2 fans only come on when the outside temp is 90-100 and I'm in stop and go traffic with the A/C working.
it sounds like a thermostat problem, How long does your car take to warm up if it just idles from a cold start. It should go from 30-40 degrees F to 180-190 degrees F within 4-5 minutes. This should be the middle of the gauge and that means the the heater should blow very warm at this point. If not then the thermostat is getting 'lazy'. It may even set a Check Engine Light for 'slow warm up'.
If the Check Engine Light is flashing it means that there is a misfire. The flashing CEL is to warn you that the misfire is severe enough to damage your CAT converter. Be prudent with how much you drive your vehicle in this state. Misfires are most commonly caused by worn out plugs, wires or bad coils.
Do a static timing check by lining up the cam and crank on the top dead center marks and then pull off your distributor cap and see if the rotor is pointing directly at #1 in the firing order. if the cam and crank are perfectly in time with their marks and the distributor is off, then your need to reinstall the distributor as Andy suggests.
It is unlikely that the ambient temp sensor would cause this problem. Check your coolant level. Low coolant will cause inefficient operation of the heating system because the heater core has little or nothing flowing through it so no heat gets transferred to the passenger compartment. Does your heater work normally at higher engine speeds?