My Two Cents
Hello qqwwweeee1
If the landlord hasn't meet the terms of the contract and the lack of repairs is that much of a concern to you, you can push the issue to get the results you want.
However, based upon the remaining time left on the current lease, your future desire to renew that lease and maintaining any good relations with the landlord all should be taken into consideration prior to any action taken on your part.
You may hire a professional door hanger to install a replacement door of fair and reasonable value and labor costs, then deduct that exact amount from one months rent, since it is stated in the terms of the lease. Your logical reason for doing so can be stated as a safety issue since the door cannot be shut nor locked.
The above may or may not get the desired results intended. You'll get the door installed sure, by the costs in future relationship with the owner may bare a heavy price with any future dealings between parties.
The sink is a losing issue. The decision is up to you now.
Case closed...Court dissmissed...