The cutting method is pretty standard. It reproduce both sexually and asexually, but in this hobby, the latter is what we see. Some reefers do see sexual reproduction, but it is more rare for most desireable type of mushroom.
Well you could do that... but the thing is that mushroom corals can reproduce super fast if conditions are adequate. If yours aren't doing the thing, check your water params out. Maybe it's lacking in something, or has too much of something else? Shrooms can cover a chunk of rock in a very short time and you shouldn't have to manually propagate them, no?
I find they grow too fast and would be more concerned about them spreading then about propagating them. The simpliest form of propgration for mushrooms is to lay a few loose rocks around the colony and let the colony spread out onto them on its own. It takes a very short time to have these small rocks covered with shrooms, after which you can remove the rocks and replace them with new ones.