If this is a rental you could just fill the cracks and redecorate. If the roof is letting in water you will need to get it sorted. My mum has a rental in Swindon with subsidence cracks. Five years on and not much progress with the insurers.
You can either rip it out or just leave it. You may be able to leave it for ages without it being an issue. As an interim measure, you could aco round the edges as you don't want any water build up in that corner. You can get cheap stuff at about £12/m and dear stuff at £40/m. Easy to fit.
Is the earth under the joists damp? It needs to be completely dry. Is there any ventilation to outside? Ideally there should be airbricks either side and a crossflow of fresh air. Can you rip out the timber floor, insulation and screed, or is that too disruptive?
In the past I have put in temporary floor decks with 18mm T&G OSB which was later binned and replaced. This was with treated timber joists. As it was a 3 storey building and a bit fiddly there was no chance of getting up and in the dry quick enough.
I have seen mist systems specified in student hmo kitchens. Expensive but effective if you only need to cover one room. For whole house cover a sprinkler system is far more cost effective.
Brickwork looks the nuts. DPC / gas membrane anal but in a nice way.
I bought sand from Cornwall for the colour. £80 a pallet haulage but the alternative was awful.
If you are not keen gardeners the greenhouse will be unused. A shed for the mower, strimmer, hosepipe and other garden tools and furniture may be more useful. The Griselinia hedge may be a bit of a solid green mass. Maybe rural hedge mix of, say dog rose, hazel, dogwood, hawthorn would be more wildlife friendly and give more seasonal variation.