Manure rots down fairly quickly. In about 6 months time you will have to top it up. Fairly fresh manure might be too rich/strong for the plants and burn them.
Just took the side wall off our built in wardrobe. It was next to an uninsulated cavity wall. Clothes started to have a musty smell and felt clammy. The end board was coming off the wall in one area.
Surprisingly there was hardly any mould just a patch 2 by 2 cm. Most of the plaster is covered...
I don't think you need to turn the compost every week. You can still put more compostable material on top of the card board. We add to the compost all the time with layers of card in between.
You need patience with your compost heap. Of course on gardening programs they show you perfect compost done in no time.
In real life it takes about a year to get good compost.
Have you googled 'hot composting'?
The freeholder has to provide several estimates. Have you got something like a tenants association/group? Speak to some of the others living in the building and try to have a meeting with the freeholder/maintenance company.
Have you considered using second hand double glazed window units instead of sheeting material? Have a look in skips or ask your local glazing installers. On our allotment site there are some lovely greenhouses build that way (by handy people).