City cops have every responsibility to uphold the law on county property. You are either ill informed or making it up.
Not necessarily...it varies from state to state...
www.quickanddirtytips.com/business-career/legal/police-jurisdiction
The short answer is that generally a police officer only has jurisdiction within his city limits, but several exceptions apply that might give the officer jurisdiction in a specific case. Also, some states give city police jurisdiction within their county, or even in the entire state.