Nice trip report and re-affirmed my belief that I have a lifetime of climbing in my backyard and no need to venture to strange lands where you can't understand the speakers at the airport.
The Alps are super fun. My wife and I climbed the Italian arete and traversed the Matterhorn, descending the Hornli (Swiss side.) Maybe 15 years ago. A great adventure. I found that a few well placed, 'accidentally' dislodged stones eased the scrimmage for the fixed lines substantially. I also enjoyed trudging down through the ski area on the Swiss side, skiiers slaloming around us. This was before snowboarding took off, so nobody ran us over. :-)
The raps are NOT fixed. Even though there are a plethora of fixed lines, climbing the Matterhorn still requires you to employ those handy mountaineering techniques (mountaineer's coil), roped travel, etc.
The fixed lines are more to facilitate ascents by the guided parties. There are anchors wherever rapping was a desirable descent option. We used an 8mm 60m rope. Any of these should do.