Cabinets, doors and blinds, and all that dust in the vents will also hold that odor.

Stale odors are probably not going to cause emphasima, but actual smoke might...

First several attempts to remove odor with enzyme cleaner have failed. 

Are there noisy neighbors, a busy street, a smoking odor in the house?

As soon as I walked in I noticed a terrible odor.

I also inspect regularly at beginning (for any random reason) to see if any problems are apparent.cats are strictly prohibited due to potential odor problems that are very difficult to resolve. 

If you have carpet, you have to worry about odors or the dog digging it up.

I would be surprised if there is not pet odor given one dog and two cats.Wait to mail back the deposit until the last say allowable by law in your state.

Once we got most of the smoke smell out then the hidden pet odors were noticeable. 

However, it can certainly move odor.