Told my realtor in descending order the deals to offer and our price, with a 3 day offer subject to inspection. 

What's really being asked here is should peoples decisions be policed? 

Can I ask the police to check for something?

Give them your business card with phone number and tell them that if you see anything suspicious call the police and you no matter how late or early.Gray area stuff, the home isn't yours and you could be in hot water if the listing agent or seller find out that you placed a cage over the unit before you close so tread very lightly.   

Neighbors called Police  and insurance adjuster came out.

Vacant - doesn't equal permission to access; the neighbors call the police everyday as vacant properties really are a nuisance.  

When the name that came back to the doctor didn't match the name the patient had given, the doctor called the police, and the jig was up.  

We called the police and they were all arrested. 

In the event that it looks vacant, I close, and the squatter breaks back in (while I’m at work or having it tented for termites, etc) and claims they live there, how do I ensure that I have the right documentation where I can call the police to take care of removing them?

But this can change if there is a police report and Code Enforcement is requested to perform another inspection to get rid of anyone the city "thinks" is a problem resident or landlord.Residents with special needs (veterans, deaf, mute etc.)