Do you mean cut the head off the base? No problem!
Do you mean cut the entire thing down to the rock? Probably!
I've heard many stories about people having their toadstool ripped up or knocked off, and it grew back. The harder part is getting the frags to stick to the new plug/rock. I don't think I'd do it on purpose though. A bit risky in my mind, but you should have plenty of frags to replace it with if it doesn't grow back.
you can but it will release a lot of toxins and the coral will be stressed and shrunken up for a couple days. i would add carbon and do a small water change depending how big your tank and toadstool are. then you will have to place it somewhere to attach which will also take some time