The Zuriberg trails are easier then the Uetliberg trails. I would say they are possible to do on a rigid bike providing you are using off-road tyres and have decent enough wheels that will take a few hits. I think trekking bikes might be a bit out of their league, but any mountain bike should be OK.

In terms of biking ability, you just need to take a sense of adventure. You might decide to walk one or two sections, but the trails are not too challenging mostly, and difficulty is controlled by how hast you chose to ride the trail.