If the kids sleep with their doors shut upgrading their doors to solid core may help. Barn doors aren’t always great a noise prevention but if you want to go ahead use a solid do and have the top of the current cavity boxed in.

You may need to add an architrave around door frame to level with the skirting. Run arch across top of door and box it in as a dropped header . You will reduce noise and provide privacy, with a sliding door but not as well as a hinged door as said

I would have the Barn door in the passageway that way you wouldn't see the pelmet from the dining room, keeping the cornice in one piece on that side.

You could easily put a door hinged on the right. Tho I am not sure what it is by the shelving there: would the door be in the way? The other very messy option would be to put a cavity slider into the wall. the section of wall by the table would all be wrecked but a good builder and plasterer would be able to finish it well so you cannot tell it has been added later. That to me would be the best option: messiest for but long term. Bard door will need a track across the top and there is not the space for that without rebuilding the wall too. and I doubt they will stop noise traveling anyway... do not want to go to all the trouble and find it makes no difference! Cheers go well! let us see what you do!! Margot