I believe that your provider sends a series of clicks (stutter) down your line to trigger the 1571 light on your phone if there is a message waiting for you. Interference on your line could appear to be stutter so could trigger the 1571 'Message Waiting' warning. That's why I ask about your microfilter, if not working properly, the broadband part of the signal on your landline could be interfering with the signals to your phone (it's just a guess and could be a red herring). Also, Marydoll is right to suggest moving your handset further from your PC, your PC could be causing the interference.
It's most unlikely that your landline has been hacked - it uses completely different technology to a computer.
I hope you get to the bottom of these problems - they are always tricky!
Searching for recipe in Supercook magazine