We are hoping to make in excess of $50,000 rental income per year so the costs are negligible in comparison with that.

I have a clause in my lease that states that negligence on the part of the tenant that causes damage to an appliance (or the toilet/sink/shower), is their responsibility.

I certainly would not want a nightmare with the first property that I buy in my local area and would absolutely not want to injure a child because of my negligence.  

However, I'm now a Florida (Pinellas County) landlord with a Property Manager in place and they are telling me that under Florida law, the tenant can't be charged for clearing drains unless some negligence is shown or proven...does this sound right?    

Repairmen assess that that water from the fridge has damaged all of the hardwood under the fridge to the tune of $1400 or so.In this situation, I would assume it is pretty straight forward that their negligence in getting the repair guy in or properly defrosting the freezer to minimize water damage is the cause and that I could take that out of the security deposit.

I don't want to be negligent and NOT send out somebody when the situation warrants, but I also don't want to pay for unnecessary service trips. 

However, insurance is limited because it only protects you from one type of liability: accidents/negligence.

I wouldn't worry about the tenant/owner below you suing you because the burden to prove that you (or your tenant) were somehow negligent would be 100% on them to prove. 

However if there are large dead branches that clearly should be pruned out of the tree before the damage happens, then you as the tree owner are responsible because it is negligence and not “act of God.”

If this upcoming water claim is something that could have been prevented and was due to negligence, sometimes a property insurance company will punish you for this with a rate increase.