---So, ask about their standard operating procedures and make sure your management contract has an easy termination clause.What are your options if you experience one of the above?

Personally I'd want to know any expenses over a specific dollar amount and would want to know about lease violations, rent that wasn't collected by the 15th, tenant requesting early termination etc. 

  $5 for batteries and whatever your time costs. 

The lease terminates on June 30 but I'm thinking of starting the eviction process as they havent paid rent and is not responding to communications.

Yes, the law provides for legal reasons to terminate the lease.

Would you renew her lease, or terminate?

One is the contract termination one page form, and then release of earnest money. 

The current landlord is giving lease termination notice before I close I plan to rent the upstairs apartment to current or new tenant.Do I need an umbrella policy or will homeowners insurance be sufficient?

Quote from @Art Maydan: I don't know what your contract says, but you should have the right to terminate if their is a breach in the agreement.

Breaking or terminating a lease agreement early can be a headache to the landlord.