Moreover, you should also consider any negligence in repairs - you want to tighten up the screws because you're already bleeding most of your income to your creditors. 

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

Hum, I’m wondering, should I save some printed copy of the newspaper to negligently post in the future on the door of some of my late paying tenants... :-)

I would never put anyone in danger due to ignorance or negligence and these people know that.

She's also a single mom working a low income job and in general I want to be as fair as possible -but that's a two way street and I don't think its fair for me to bear the cost of all of these items that were likely due to her negligence/that of her kids.  

My leases don't have rules about hanging things outside, but they do say tenant is responsible for negligence or damage.

Run down properties do not always mean a negligent landlord.If you get a chance to talk to the long term tenants, ask them what they like about living there.  

Any lawyer will tell you that ultimately if you do something grossly negligent, malicious, etc it doesnt matter if you have an LLC or not - its bad news.

This change is going to cost me $31/mo for the life of the loan (should I go forward with it).Anybody think there's a chance I have a claim against them based on negligence or anything else?

Could set yourself up for willful or negligent misrepresentation or omission if you view and they hate the place.