And then, if there's an infestation of ants all of a sudden, etc., they usually will treat them for no extra charge, as they include them if you get a monthly service plan.7) Utilities paid by you?

Material affect on health and safety of occupants (I) infestation of insects, vermin or rodentsI called the day after to try and get out of the contract due to the infestation.

An infestation might lead to a health hazard and, who knows, they may send an official to check and see whether the issue is hazardous, forcing the owner to handle it.Good luck, that’s a very nasty problem and it seems as though you’ve handled it to the best you could on your own.

The tenant has access to backyard (not in lease but offered as a freebie) it's in decent shape with not trees or plants whatsoever but my neighbor's back yard is lush green and some of his plants creep into the backyard.There will be occasional uninvited guests such as Centepedes and Beatles when the tenant opens his door.Now they are asking me to call in for an exterminator,the previous tenant never once mentioned it to me.Its not an infestation so I think it doesn't require me to call for an exterminator,it's not an infestation per se.How do I handle this demanding tenant?

Anyways it's a duplex and the apartment is infested with roaches and every other insect there is.

This way, no more infestations and our bug guy gives us reports after he has been in the property.  

Michael Seeker has a pretty good point, even if it is their fault, you still don't want a property infested with anything.

I think it's best for the LL to do it, because it protects your investment, and you could be on the hook for habitability issues if the tenant wants to claim you are running a slum, etc.In CA anyway, if you hire a pest control service that you pay every month, they come and spray every month and they will also handle any infestations that happen without any extra fee.  

Property shall be free of insect and/or rodent infestation;10. 

This is the first month that there excuse holds any weight, the home became infested with bedbugs which cost them around 600$ to treat.