Lots and lots of advice here Ash - a simple truth in these situations is that you often have to remove the tiles as carefully as possible then assess what (if any) repairs need to be made before starting tiling.
If you need to box in something that has to be accessible, Silicon is a good option, but you can also utilise these magnetic catches
Magnetic Catches - 4 pack £4.92 in stock - next day UK delivery
There are a lot of options when repairing plasterboard after tile removal. if holes/damaged plasterboard is less than 25% of the tile area then you may get away with tiling straight over the damage, although you need to assess - rigidity of remaining plaster board, thickness of tile, security of the wall area as a whole.
IMHO remove the tiles in any event - folded dust sheets on worktops will prevent damage to surfaces. Assess walls, repair if necessary and re-tile.