I cut all my patches "at the muzzle," whether actually the muzzle of the rifle or as I fill a loading block.
As for the patch knives themselves, there's only one rule for me: They have to be the sharpest knife you own, and then some. For best results you want to cut the patch in a single swipe, and that takes a really sharp knife.
It's also moved me away from little dinky knives. Short blades are hard to cut patches completely and easily even when they're sharp enough. I've settle on a couple of patch knives now- one has a 4" blade and is almost too short for my tastes, and the other is just right at 5".
As for carrying them, you name it and I've probably tried it- around the neck, on my belt, in a scabbard sewed onto my bag strap, to the end panel of a bag and on the back of the bag. Backed off all of those for one reason or another. Mostly we hunt in dense brush and anything dangly- whether attached to a bag, belt, neck or strap- Tends to hang on the brush. Hate that! I carry a small day horn inside my bag rather than dangling for the same reason.
The carry that turned out best for me is to keep the knife in a looped belt sheath in case I want it on my belt at some time for some reason, but mostly it gets dropped sheath and all into my bag. I have a lotta bags (one for each gun), but the patch knife is so critical for my tastes it migrates between bags, rather than having a knife for each bag.