I'm not familiar with this particular model, but I am familiar with a few of the other model MB Climate Control systems. The following assumes that yours operates on the same basic principles.
You asked where to start and I would say, in this case, to check your fuses first. In the systems that I'm familiar with the pushbutton control unit is the heart of the system. When you have multiple problems such as you describe, the problem can be related to the pushbutton control unit. If the fuses are good, you will need some detailed information about your system so that you can start by checking for power into this unit.
Also, once you ensure that the fuses are good, push every button on the control unit to see if you can get anything at all to happen. If everything seems totally dead, follow the procedure in the manual to ensure that the pushbutton unit is bad before buying one. If you don't have a manual the CD would be your best bet.
If the troubleshooting procedure does indeed verify that the pushbutton unit is bad, MAKE SURE that you test the auxilliary water pump for current draw. If it draws too much current, which will be indicated in the instructions that come with the new pushbutton unit, replace the pump. If the pump is drawing too much current, it will blow your new unit. The pumps usually start drawing too much current because of a dirty cooling system. Frequent flushing of the cooling system, followed by fresh coolant will prevent this from happening.
Don't just buy a pushbutton unit and stick it in to see if that is the problem. At a minimum, follow the instructions and check the auxilliary pump current draw.