Those are some nice pieces. Here are my suggestions:
1) Don't remove anything from any of the discs/plugs
2) Just superglue the flat discs onto target spots unless they're tricky spots (see my previous reply)
3) Assuming photo 3 is Xenia and photo 4 is GSP, I'd strongly suggest mounting those on "island" rocks. Those two corals grow super fast and can quickly take over any rock work they come in contact with. Using isolated island rocks will keep them mostly isolated (although Xenia in particular can break off and sneakily create colonies elsewhere). Frankly, I'd just toss the Xenia frag in the trash---but that's just me because I hate Xenia and it totally overran one of my tanks.
4) The trumpet is on a plug. If that's the spot where you want to place it, snip the plug stem off as close as you can to the disc using a pair of heavy wire cutters, tin snip, etc. If the disc will fit nicely in that spot, just superglue. Otherwise, order some aquastik and use the method I described above.
Hopefully this is helpful advice. Good luck!