I am getting king of crazy (sad, frustrated, desperate,..., curious,...)
Yes it is understandable. I don't think there is a sympathy smilie.
should the Arduino code run on Processing also?
No, while the syntax is similar it is the language C in the Arduino and Java in Processing, so it will not work.
Is it possible to see WHAT is getting there?
You can add print statements to processing to look at any variables while it is running. That could give you an insight into what is happening.
I was asking a friend about this yesterday and he said he had trouble with this program at first and it was down to the way the MIDI configured using the Audio MIDI setup application. The website for the serial <> MIDI converter had a guide to setting this up have you gone through this?