Congrats on the new car. You got it for a really low price so smaller problems like these are to be expected. It's a great car, though it's not for those who want lots of acceleration.
Did you try pressing the stop lever really hard all the way? On my 240D I have to press it quite hard if I want to stop the engine this way. I'm not sure if this is adjustable. Check the vacuum line going to the back of the injection pump. That's what controls engine shutoff with the key.
As far as starting is concerned, how long are you letting the glow plugs glow before starting the engine? A good 15 second glow is what it sometimes takes to get a smooth start and idle (ignore the glow light). You can also check the glow plug resistance with an ohm-meter (do a search on this board). I would also do the fuel & air filters, and valve adjustment. Don't do any compression testing until then, otherwise you'll get inaccurate results.