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If the battery is completely discharged a couple times then it should be replaced. I just finished dealing with this on my 318i. It was an Interstate battery w/ 2 years left. That's the 2nd Interstate that died prematurely. I've had great luck with the Autozone batteries. Had a 7-year last for 8 in my VW.
I'm not exactly sure what the reset socket on your car looks like but if you buy a Bentley manual on it it will tell you if you need the special reset tool or if you can just jumper 2 pins to reset the service lights.
Your tranny could have 1 or 2 problems. Syncros are going out. Shift into second before going into 1st or reverse. And sometimes letting the car roll a little will help. Also if your shift bushings/mounts are worn that will cause problems. If your shifter hits the console when you put it in reverse then the linkage is worn. I just did this job on my 318 and it's almost identical to an e28. The parts are cheap from the dealer but the exhaust and drive shaft have to come off to fix it. While you're there might as well replace the flex disc, center support bearing and tranny mounts.
You are on the right track on the cold start problem. Again, the Bentley manual will walk you through the checks. I would run some BG 44K through the intake manifold to clean the valves before doing the checks.
You can do it yourself or pay a garage alot of money to do it. They have a special fogger they attach to the throttlebody that does a good job.
If/when you do this just post or pm me and I'll give you instructions.
Good luck with it. Sounds like you got a beauty. I really like the bottom end torque on my 528e but I lust after my dad's 88 535i. Fast!
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