In the process, a door was broken, and the tenant responsible for that property damage (herein referred to as the "problem tenant") moved out.

Kudos for the DIY effort but once you're prying out screws (maybe someone repaired it before and stripped them) you're doing some damage and probably just begging for a call with another issue in a couple of months.

They never notified us and the spoon caused the disposal to deteriorate and leak which caused severe water damage to the cabinets underneath.  

Steven Taylor I never, ever let tenants do pool care; chemicals not right, damage pool finish etc.... take care of it professionally People share pools all the time in apartment communities, so I don't see a reason why it can't be for both houses.

We inspected the unit and noticed damages of over $1,000.00.

One day a tenant will move out and do damage that will cost you $5000 and a month of vacancy.

A good rule of thumb to remember with insurance claims is sudden and accidental,  such as a tree falling on it, or wind damage, if it is something that happens over time your insurance carrier will probably deny the claim.  

Some other issues to consider that could and have happened to owners, is that the property could be damaged by a hurricane, flood etc... and they have spent all of the money, which essentially needed to be refunded to the tenant.

If you want to get water backup I would yourself "if it was to occur, what would be damaged?"

It can be to their financial benefit when the tenant causes more damage, where it is never a benefit when you actually own.