John, txs for referring to the "discussion on mats" post: it really gave me good ideas.
Cleaning the slits seems OK: I use a hard brush, applying it lengthwise over the slits. This seems to work easier than poking the holes clean.
Turning using the larger bamboo mats works quite well: I put the mat, with the cheese on it, in my left hand, and drape a clean mat over the top of the cheese. The top mat automatically folds down along both sides of the cheese, which then gives me a steady grip, using my right hand, to tip it over. I then gently peel of the top mat which was at the bottom before.
However, after reading the earlier mentioned "discussion on mats" post, I will look into a way to airate the cheese better.