Our new policy is to charge a commitment fee to this type of borrower if they return in the future.

If there is going to be a wraparound (or Subject to) mortgage make sure you buy a title policy through a Trust friendly (build rapport for future reference and deals) title company as to satisfy The Seven-Day Notice Requirement: Sec. 5.016(c): "where the purchaser obtains a title insurance policy insuring the transfer of title to the real property" Even then, you are still closing a deal without protection of your assets.

I am thinking about an agreement that the renter/lessor agrees to accept responsibility for the safety of themselves, family, guest, and anyone else that they cause to be on the property.

You can enforce a clause, if present, in the LC that the Vendee obtain and hold proper insurance but you would still want a policy of your own.  

Not that I depend on this, but it's another nice safety net -- but insurance on the house/my stuff is my main safeguard.I let complete strangers who I thoroughly check.

The Staging company stated that they also have insurance, but he said it would affect their policy and put them in a high risk policy so he would rather get paid in cash.

Liability: I firmly believe that every landlord should have an umbrella policy.

Than I found out it was an awful idea becauseviolated the mortgage (not a big idea now but gave mortgages a reason to come after you in a couple of years when rates were higher)no seperation- we self managed so we were unable to hide behind the llc, had we not been managing it, it would have been differentHard to get a Umbrella Policy- If it is under an llc you needed to have a cooperate policy much harder.Honestly we just have all of our houses under our personal name.

However, it is worth asking the question ahead of time to see what the bank's policy is. 

Foremost will allow you to add as many as you want to one policy, although I don't know of a major price break or discount that isn't simply just the multi-policy discount.