The putty is a standard sealant used to help reduce movement of the pipes and fold evaporator trays etc on to the compressor.
The liquid could be condensation, and getting the staining from the black pain of the compressor, however there is a risk that the coolant gas could be escaping, so I would as a matter of urgency get someone to inspect the appliance. A small leak can easily be repaired, but the re-refrigeration is an irritant so there is an urgency to get someone out sooner than later.
If you would like me to recommend a suitable engineer to come out and inspect the leak, please provide your post code.
I had a chat with a colleague, and he said it sounds like a overflowing \ cracked evaporation tray, it sounds more likely after re-reading your question, as your suggested its coming from under the motor not where the pipe comes into contact with the compressor.
I have referred customers to WRS in the past and have always had good feedback from the people who use them, they charge £45 call out, but that include the first hours labour. They cant be contacted on 01484 297004