Left picture is some kind of Favia. Right picture is a Ricordea yuma.
As for removing them off the plug, that's up to you. You can use epoxy to stick the plugs directly on the rock or take the coral off the plug and then put them where you want.
To remove them off the plug. Sometimes this can be done without damaging the coral. Sometimes it can't. It depends on the coral and how much has grown onto the plug.
Ricordeas are a soft coral, a mushroom essentially. This means they have no skeleton.
You have to be careful not to cut the flesh for ricordea. It looks like you can use bone cutters to break off the base of which the ricordea is attached to. You can't glue the tissue of ricordea but you can glue the piece of rock that its attached to. Sometimes they die if you glue their flesh directly, as with most soft corals.
So lets say you got the ricordea off the plug and you glued himomewhere. If he's stressed, he might decide to move, yeah they can move slowly, off the rock and attach somewhere else. OR he might detach and float around the tank stinging corals or getting stung and then die. Not to worry, if he detaches just put a shotglass in your tank with some rock rubble. The shotglass gives him an environment to attach to the rock rubble.
As for the Favia, you'll have to make the call on how bad you want him off the plug. If he's grown(as in lay down skeleton) onto the plug, it will be hard and you might want to break off the plug with bonecutters. Often times, I'll use a flathead screwdriver and a small hammer to chisel corals off, making sure I'm chiseling the superglue and not the coral itself. That picture looks like you could give him a little chisel or try the bone cutters.