Property uninhabitable due to tenant negligence.

Since you were not negligent for this damage, (sounds like a back up and due to no fault of yours or your tenants) your insurance would not be legally liable for the damage to the unit below.  

Named additionally insured on your policy; Hold harmless agreement; indemnification from both the property owner and tenants; Can't be held liable for mismanagement and or gross negligence; Rent checks shall be made out to the PM and will be deposited into PM's bank account; PM decides what expenses are to be paid, how much and to whom; Property owner will receive a disbursement only after the monthly financial books are reconciled; PM has a sole claim on a myriad of tenant fees; Property owner will be billed for a setup fee; Property owner will be billed for tenant dwelling inspections; If the property owner suspends the contract, there's a bill for that too.

Lease states “if any required repair is caused by the negligence of the tenant or tenants guests the tenant will be fully responsible for the cost of repair/and or replacement needed“ Should this be claimed on my insurance? 

This sounds like your PM was negligent and should be liable for some of the damages, especially if you can prove they failed to adhere to the contract.

Notwithstanding the provisions of this section, the resident is responsible for damage caused by his negligence.

In my market, change in rate between, say, 5-8% are so negligible in mortgage payment that I gladly concede that to get the 0% down terms that I like.

It is questionable whether the property owner is truly responsible for these costs because there may be some negligence by the HOA for creating this problemI would like to propose to the neighbor to enter into a contract for the sale of the property with the following conditions: $85,000 at closing but only $55,000 of this amount will be released to the seller immediately at closing.

Interest rates are negligible, but I like having these separate from my personal accounts (where I do utilize things like high yield savings accounts, etc.).

If you were negligent in causing the injuries, and if you were not remunerating your friend in any way, your policy might, depending on the circumstances.