To change to cavity doors you will need to remove the wall each side of the doors to accommodate the cavity housing. Normal sliding doors would make a difference but the problem with them is they never stay totally closed and they do become quite noisy and annoying. This problem also applies to double cavity doors. The other option is to use bi-fold doors, they won't intrude into the room as much. Do you really need two doors there at all and is the glass above the doorway necessary? My personal preference would be to fill it all in and just have the one door to the room. You could even re-use one of your doors.

A cavity sliding door will be the solution to the shelf problems. But in any situation it requires to be built into the wall. Brick wall is I presume a bit more work. A wall mounted sliding door will take up space on the wall. It's just about working out where is best. Need the wall shelves in the ensuite? Then it's outside, and pull the furniture out from the wall a little so it can run along the wall. Really need a bookcase or other wall unit right on that wall? Consider if you can move the bathroom shelves. :)

FormFunctionNT I love your lego door! So innovative. This idea may be a little busy for the other door, but just putting it out there... You fill it with squares of the 4 colours, but randomly leave 4 blank. Almost like 4 have left and moved 'next door' if you'll pardon the pun! Sorry, I couldn't help myself.

Is the door solid core? If not, a solid door would help somewhat, but would have to check the track can withstand extra weight. We have a couple of very heavy solid hardwood doors in steel frames which run on steel tracks. The bignheavy doors do dampen noise but certainly doesn't eliminate it entirely. Heavy drapery, rugs, soft furnishings, tapestries and large textile wall hangings will help to absorb sound from within the room, but still won't entirely eliminate it. If you don't already have carpet runners or rugs in your kitchen, adding some will also will help. Same goes for any other rooms which adjoin the music room/library. I knew a couple who soundproofed their small music room when they built their house. Their son played drums. As I recall, they used a special acoustic plasterboard for the walls, and the floor was carpeted. I think it helped considerably. there are also acoustic panels/tiles that can be installed on the ceiling, but this may be more trouble than it's worth. I'm not sure but the installation might be something like a suspended ceiling.