The neighbor has to file with his homeowner policy.

Others lean toward simplicity and think you should just get a good insurance policy and put them under the same LLC.

Lead, Mold and Structural issues can all be excluded on insurance policies depending on the claim scenario. 

Also we have a large umbrella policy that covers all of our houses.We also own two separate LLCs - a transport company and an espresso stand. 

Some, like State Farm, like their properties to be in good condition before they'll issue a policy.

That being said, there is something to say for having someone who knows "how" to present a claim relative to a policy.  

I saved about 1000 dollars on my renewal for my policies because he switched me to a new provider due to my request.  

good luck... sounds like you have a pretty high bar to get across.sinkhole is  an insured loss  FYI I have funded well over 150 deals In Orlando last 3 years never lost a one.. but in reading our insurance policies if one got swallowed up then the insurance company owns it.there is no Utopia in rental house investing..

Also, do insurance companies offer umbrella (blanket) insurance policy for multi-state properties?

Yes they can, but it is not the only reason they could offer a higher premium on the renewal.If your area has had a lot storm related claims, like wide spread hail storms, chances are all policy holders in your area will see an increase in premiums.