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Is the liftgate rusting or is it just the trim piece? Maybe the paint was scratched the last time it was fixed and was susceptible to rusting. Rust or corrosion should definitely be repaired before it gets too bad
Jet has instructions on their website to use their programmer. http://www.jetchip.com/howto.asp?folder=FordPerformanceProgrammers You will need to copy/paste that link. The programmers plug into the Diagnostic port, most likely somewhere under the steering wheel on the lower dash. I think the PCM is in the engine compartment near the battery. You can probably find some good info about the Jetchips from a Mountaineer/Explorer forum. Many people make good how-to write-ups on mods
Is the battery old? It's possible the battery has become weak and this cold weather really drained it. Cold weather really does a number on batteries. Most shops can test the battery for you and many auto parts stores will test it for free if they have the tool. It's possible something else is draining the battery when the key is off but that would require some basic electrical knowledge and a multimeter to diagnose
I'm not familiar with your Mercedes but the people at Benzworld may be able to help with your question. http://www.benzworld.org/forums/ If you have an S420, that would be the W140 chassis (known internally as the 140) http://www.benzworld.org/forums/w140-s-class/ They seem to have a stickied post at the top containing user manuals. You'll need to copy/paste that second link for it to work properly. Hope this helps!
Is there any apparent leak that you can see? Do you see white smoke from the exhaust? Check your oil dipstick for oil and coolant intermix. It will look like a mayonnaise that has turned. Does the coolant smell like exhaust or gasoline? If the car has overheated, there is a good chance the head gasket has blown but it should be verified.
You can check out Autozone's website, they offer free repair guides online or maybe consider buying a repair manual from the local auto parts store. From the manuals, it looks like the alternator is mounted down low and some things need to be removed to gain access. I've found that online forums can be really helpful with how-to write-ups
For many repairs, Parts and labor at many dealers has a 1 year warranty attached to that repair. If the dealer has just replaced the transmission, and it's already whining, definitely call them
You can also check the oil dipstick for oil/coolant intermix. It will look like a mayonnaise that has turned. You can also look on the underside of the oil filler cap for evidence of intermix. Why do you think the head gasket has blown? Is the car overheating? Are you seeing white smoke from the exhaust? Head gasket problems were not uncommon with the 3.8L
Something in the shift linkage may have broken if you can shift it manually under the under hood.
A defective fuel pump will rarely set any trouble codes but is a likely culprit for crank-no start conditions. You'll need to check fuel pressure. Try to listen for the fuel pump when you first key on (it will automatically shut off after a couple seconds). Also look for any blown fuses.