Sorry you are having problems.
Not totally sure what the issue is there. Presumably you are using Network MIDI? In which case all the networking side is taken care of by OSX/iOS and Pianist Pro merely asks iOS to create a connection. I don't see how the port used by Network MIDI would conflict with standard web ports. And you say that you disable all the web server applications and Network MIDI is still giving you problems? Which sounds to me that there may be something else that is going on.
Also you have your DAW running, and although I can't think of a reason (non-intentional pun) why Reason would have trouble, I do not own that specific DAW to test with.
What I'd do first of all is to disable / close all these apps and just run MIDI Monitor and then connect to Pianist Pro via Audio MIDI Setup. If you see the correct MIDI messages appearing in MIDI Monitor then that proves the Pianist Pro--Network MIDI connection, and you know that it's the path from there to your DAW that has issues.
Also, one last (stupid) question: have you checked that you only have MIDI OUT set in Pianist Pro and not MIDI IN also, since some DAWs seem to echo incoming MIDI back out again, and Pianist Pro can also do that, causing MIDI feedback.