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.

So, hire a pest management company to deal with your unit.  

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?

Sounds like it's time to call a pest control company. 

Having said that Minnesota does not have termites due to our weather so we don't do pest inspections, we don't require hurricane insurance, or earthquake coverage, but many but not all lake properties require flood insurance. 

I 100% know that my desires (example: I'll happily pay for quarterly pest control) aren't the desires of all of my PMs clients (they don't). 

fixable and treatable. a pest control professional and advise you if they are still active. 

And an infestation of German roaches that made the pest guy jump (he said).After everything was said and done, we incurred 4 times the cost as I was told/ estimated. 

Hello fellow homebuyers,I recently made an offer on a house with an inspection contingency covering the roof, foundation, electrical, plumbing, backyard fencing, and pest infestation.

On top of that I had the city coming after me for fines due to the condition of the place   This was a drug infested area of town.