I have been thinking about a trip to this area since you recommended it after your last trip. I bought a hiking guide for the area - Cicerone Guide - Walks in the Cathar Region.
The map in the book shows that the GR77, French long distance path, passes right through Minerve.
This page shows the map of the trail in that area. If you could get more information about the GR77, an option might be to walk parts of it in the area. The GR trails are usually well marked.
The Cicerone Guide has two walks from Minerve.
Walk 1 - 18km/11.2miles, 5 hours, 180m to 400m (220m gain)
This walk follow some of the GR77 and is a circular walk, so you get back to where you started. You see a dolmen on this walk.
Walk 2 - 6km/3.7miles, 2 hours, 150m to 340m (190m gain)
The other towns covered in this guide are Beziers, Lastours, Foix, Roquefixade, Lordat, Montaillou, Puivert, Montsegur, Usson, Puilaurens, Queribus, Peyrepertuse, Villerouge-Termenes, Rennes-le-Chateu, The Pech de Bugarach. They have long and short walks for several places. I have found these walking guides to be very good. We used one for the walks we did on the Amalfi Coast. I have the one for the Cotswolds (England) and it is good.
A guide for us is that a 300m gain is not bad. 500m is doable. We try to avoid steep downhills (hard on the knees).
Another thing to look for, and I will do some searching now since I am interested in this area too, is to find walking maps put out by the department. The maps published by Gard for the area around Uzes (we have hiked there a few times) are fantastic. Good maps, descriptions in English, good hikes.
It would be fun to seek out many of the Cathar Castles. I have not been to any of them.
This website lists them all.