The problem is that often claims are submitted by the injured individual's insurance company not by the individual. 

If someone injures themselves and sues, they will be suing the LLC and not you personally.

I don’t want to sound snarky, but to answer your question, that’s the messed up part.Some people think that the mentally ill should be given a free pass for their behavior, no matter what it is or who they injure.   

Should I file a police report if my tenant was injured from a fall after being reckless?

basically anyone doing this type of work on your property should be covered by workmen s comp insurance.. if he gets injured, falls, he'll sue and your going to have issues,, better to hire a service that does it on a regular basis or get workmen s comp added to your policy for this.

Let's say he's injured while cutting the grass or shoveling the snow.  

Were there to be a break-in at your rental, this fact would certainly come out at trial should an injured tenant sue you. 

Do you worry about getting sued if a guest gets injured on one of your properties?  

Medical payments coverage is a no-fault coverage for the medical bills of someone injured on the premises.  

Here in CT, the issue that comes up with General Contractors is that, if an employee of a sub is injured, the claim works its way up the line and can impact the GC.