Sounds like a fun project.
I came very close to buying a 2001 SLK 230 when buying my C Class. The SLK is a great car.
I can't help you with much, but I can tell you what the button showing the car on a towhook is for. Because of the Electronic Stability Program(ESP) you must disable the ESP and the burglar alarm by pressing this button when towing the vehicle.
The climate control problem could be about anything, but I would check the hotwater valve to the heater. In many of the earlier climate control systems, when having the problems you describe, the monovalve would be involved. Are you sure it is connected properly?
Hope this is helpful,
'01 C Class, Six Speed
'84 Euro 240D, manual, 533K miles
'88 300E 5 Speed
'81 300D Daughter's Car
Over 800,000 miles in