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If it's not already included then yes, if it's private rental. If not he'd have to go through the real estate and amend the lease for the extra amount
If there was no heating/cooling when you signed the lease then he is not under any obligation to install it now. You were aware there was none, and were happy to sign the lease with that knowledge. If there was heating/cooling when you signed the lease, and it's broken down, then yes he's obligated to fix it.
It depends. Was there heating and cooling In your residence when you signed your lease? If so, he is responsible for keeping it in order. He can’t however increase your rent until you sign a new lease, and that can’t happen until your current one expires.
If it was advertised as such, yes he is responsible for fixing it. He can’t increase your rent until the next lease.
Only in a new contract he can't change the existing terms of contract. So at next lease renewal yes.