The answer to that is a definite yes. Phoenix is known for many things but pest immunity is not one of them. The exterminator deals with the problem of pets once and for all. You can attempt to do the firefighting route but that is not going to take you very far. It is better to just get on with it and deal with the pest problem. The other considerations can come later on.
Get a good exterminator
It is better to go for a good quality product that gets rid of all those frightful creatures rather than trying to save money. Homes are never comfortable if you have the suspicion that there is some creepy thing waiting to pounce when you turn your back. If you are not looking for a constant Halloween experience, then it is probably a great idea to get rid of termites.
You also need to be particularly careful about rats. They are associated with filth so at least you know how to prevent them from gaining traction. As a rule of thumb, light tends to get rid of unwanted pests. They prefer dark and moist places. They also like food so you need to starve them until they vacate your home.