I'm not sure if it differs between areas, but my LA and HAs have a 'lettable standard' policy that properties should meet. Google yours and see if anything comes up, and report anything that hasn't been met. I'm certain things like in your OP would be things that need to be rectified, especially the spikes in the doorway, it should be safe as a minimum.
This might not help as I say, not sure if LAs differ but the in the places I've worked some examples of what's covered and what isn't -
Things like cracks in walls/ceilings might be classed as decorative so tenants responsibility (one HA I worked at for e.g would say if the cracks were no thicker than a £1 coin, they weren't covered by the repair policy).
Painting iirc is in general tenants responsibility too, unless walls are affected by mould. Depending on the area covered, eg if its just in a corner it might just be cleaned off and stainblocked.
Cleanliness would also be dependent of the state of the property - a bit mucky or dusty would be ok, but if completely filthy, it should be done by them.
Gardens are generally tenants responsibility too BUT they should be let as usable and safe. Yours would be cleared for example.
I can't remember if there's a maximum length of time it can be reported under the lettable standard but the sooner the better, because the longer you wait it will be implied you've accepted the issues, or they will try to argue there's no way to prove if you did it or not. Take pictures of it all, too.