I've cut a bunch of mushrooms, to propagate them, and i've always had good results, there are two ways you can do it.
The first being to just cut the head of the mushroom off of the stalk with a razor blade, you can then let the head float around and attach where it wants to or you can use some mesh to secure it to a rock where you want it untill it attaches itself, and it will start to grow.
The base of the mushroom (where you cut the head off) will also start to grow a new mushroom, this is in my experience the fastest way to get two new mushrooms, the stalk will grow a new head really fast.
Now the second way is a little slower but it will give you way more mushrooms, basically you do the same as in the first, by cutting the head off of the mushroom. Then you take the mushroom head (i put it on a little plate) and slice it down the middle, just like a pizza, making little "mushroom pizza slices" make sure that you get some of the mouth of the mushroom in each slice (this will help it to eat and grow). The size of the mushroom will dicktate how many "slices" you can get out of it. I have gotten as many as ten from a mushroom the size of a half dollar (before being taken out of the water, as they will shrink when taken out of the water)
Now you can use some mesh as in the first example or you can do what I do and just throw them back in the tank and let them find thier own spot. (I don't mind them being all over the tank) I have also heard that you can super glue them to a rock and they will do fine but have never done so myself.
Now with the second way a single new mushroom will also grow from the stalk, as for the little "slices" they will be mis shaped for about a month or so until they heal and turn into regular round mushrooms
As far as hairy mushrooms go Im not too sure about those, I think they split on thier own, meaning they actually divided from one into two mushrooms.
The mushrooms I normally cut are the kind that when they move naturally they will leave a little piece of themselves behind and then that will grow into a new mushroom
hope this helps, good luck, but i don't think you will need it, they are near imppossible to kill .