Which professional should I call for leak coming down the furnace exhaust flue?
This is a common problem in windy areas with rain. In many cases the leak is rain getting inside the pipe and running down. Based on your photo I would say this has been happening for a long time.
I would want to call a general contractor that also does roofing because the roofing only guys may not be the best at repairing the interior damage. The cleanout looks like it is rusting through.  Water has probably been sitting in there a long time.