I would not install anything, this is something that is a disaster waiting for a place to happen, I am pretty sure your council would not approve a trap door in this situation, no way in case of fire this could be opened in a hurry in the middle of the night. I think your daughter will have to live with it the way it is.

I don't know much about grout but look at Zinsser Permawhite for your paint on walls and woodwork. It is supposed to resist mould on the paint film for 7 years. I have only just tried it for the first time the other day and it gives a nice finish... time will tell if it lives up to its claim but I have had very good experiences with their other products. Victoria is spot on, extraction is key, and also with your kind of building dealing with damp from the walls initially that could cause problems. (I can't help with that bit- I am tearing my hair out over a damp cellar conversion at the moment!) Good luck to you, sounds like a fantastic project!

Digging down significant affects the cost but it sounds necessary to make the room good enough to add value to the sale price of the house. Digging down likely means underpinning- therefore the conversion cost could be £3,000-5,000 per square metre plus fee, plus building a lightwell. If you dig down this is classed as an extension and therefore needs planning permission. Since this work is unlikely to affect how the building looks it is difficult to imagine a planning application would be rejected. If you are an attached property you will also have party wall considerations. If your house has no front garden all building materials and waste will have to go through your house which likely means you would have to move out. So you need to consider if a cellar conversion is worth it. Basically if you built a 4x5m room in the basement (which is the size of a bedroom plus bathroom or a reasonable reception room) you could well spend £80000-£100k. However in more expensive postcodes it might add more than you spend- the agent can better help with that but basically I would only look into it if your house is currently worth more than £500k.

Downstairs WC- I would extend the hallway a little into the kitchen area and put the wc where the kitchen sink is now. This would still leave a room that is 15’ long and big enough to plan as a kitchen diner even without a side return extension. The front reception rooms will never be bigger but if they are dark there are likely decorating and lighting solutions- posting some pictures will stimulate some thoughts from the community.