Is your deposit protected with one of the schemes? If it isn't, your landlord could have to pay up to 3 times the deposit back to you. If it is in a scheme, it isn't up to the landlord or letting agent what they deduct, the deposit scheme will adjudicate what is fair if you raise a complaint.
After 10 years, unless the property has been redecorated by the landlord regularly, it would be expected that the property needs repainting so they can't charge you for that. The same with carpets and shed, if they are 10 years old, they are unlikely to be able charge you anything as wear and tear is expected or it will be a fraction of the cost depending on the life expectancy of the item eg if the carpet was 5 years old, you have trashed it and they deem that a carpet should last 10 years, they could charge you 50% of the cost.
If you leave them, the maximum the landlord can charge is the actual cost of removing them, so the same it would cost you.
Under the current circumstances, particularly as they letting agents are difficult and the landlord is selling, I would get the 3 quotes for removal and leave the items there. The deposit scheme would only let them deduct a fair amount for removal, the quotes will be proof of a fair price. Don't agree to any deductions until you have checked your rights.
If you return the keys, they can't charge you rent just because you left a chair behind, only the cost of removing it.