Well, after exactly 2 months, I can proudly say that the machine is back :-)
It took quite a while to get the needed parts and I replaced the pulley unit with the belt and holder, as well as the encoder (which was the thing that wrongly reported a paper jam and stuff). I also ordered the Epson grease the service manual recommends for the carrier part.
Farmer, You are right, Epson recommends measuring and adjusting the belt tension with a sonic meter. I found it in the service manual, page 364 - 367. As I don't own one, I searched and found some iPhone app that provides similar functions. Sure, not as precise as an industry tool, but it does the job. You need to measure and weight the belt for it, the span is listed in the manual. I measured the old belt but I guess that's ok. The app spat out a tension of 46.6N (33 hz frequency), the manual states a value of 45N ± 3N so I hope it will be fine.
So far, the prints look good. The head survived the 2 month standing period just fine, what a relief.
Now I wonder how often I need to grease that printer....