Star polyps are really hardy. Once they have the right conditions (clean water and iodine) they should survive any cutting. The big thing in all of this is to keep the flow good so no rot can set in.
As for favites (aka close brain or moon coral) I'm sure you can propagate it, but knowing that even touching that coral can kill it, I wouldn't try. It's a stony coral and most stony corals do not tollerate much touching let alone any cutting. If you are serious about it, there's a great book out there called "Book of Coral Propagation, Volume 1: Reef Gardening for Aquarists" by Anthony Rosario Calfo. You can get it from amazon. I read through it almost every night at the LFS
I work at... It's a long book with a LOT of good information on all corals and propagation techniques, very few pictures.
I'm not sure about the Toadstool... again, I'd refer to that book on it. You may be able to do a latteral line cut like with a mushroom on it... but I'd be sure long before I went at it with a razor.
Hope that helps!