I'm not an expert on mounts so take my "learning" with a big grain of salt. My understanding is that for mounts you never remove it from a mount and re-mount in the way you would with a pot. If an orchid has outgrown its mount, the orchid on its original mount is attached to a new bigger mount.
As for all those root sticking to a pot.... I have some experience there. First soak the roots so that they as pliable as possible and then carefully remove the roots from the pot, a thin bladed knife or Xacto knife (razor blade) is used to gently and carefully detach roots from the pot. There will be some damage, but with care and patience you can keep it to a minimum.
Mounts only need to be redone when the mount rots.
As for repotting plants that have grown all over ceramic pots, the advice given by Carrie is right on. You can also simply put the pot into a larger pot and add media to the space. A lot people also smash the pot with a hammer and move the plant as they see fit, pot shards and all.
You can re mount an orchid especially if the mounted orchid was originally put on a small mount, some are if they are young plants. Orchid usually come away from the mount easily, some just place the old mount on top of new.
I like to put a good amount on sphagnum moss between mount and orchid.