By the 5th day of no rent, you should have served them with a notice to pay or quit.If they haven't paid rent for Oct, get rid of them now and try to find new tenants, but do not return the deposit until you find a new tenant. 

Maybe the seller is looking to get rid of it because he's had difficulty managing it and getting paid. 

You being able to close sooner helps make the offer stick out of the seller wants to get rid of it.

I think she wants to get rid of the property in the short term as I want to own the property as my home in the short term.  

If the Tenant leaves, the Occupant could trash the place or steal the appliances and you wouldn't have any way to prove who was responsible, leaving you to go after the Tenant while the Occupant gets away with it.What if your Tenant wants to get rid of the Occupant but the Occupant refuses to leave?

Now with a 5% down conventional loan you will still have PMI but it will be much cheaper and you can get rid of it when you hit 80% LTV. 

Get rid of her and then hire a property manager to handle things for you until you have a better grip on how to screen applicants and manage the rental.

@Gorden Lopes termites aren't like ants where you have an ongoing contract, you need to get rid of them now and treat the damage they have caused. 

Counter Depends on, how serious are you to sell/get rid of the house. 

The new tenant has found mice problem in my rental house which has recently been completely renovated. She has set up 18 traps and still sees mice activities. Any suggestions for good products? Traps, electronic repel...