The written permission said that the "guest" will be evicted if they cause problems. 

One I looked into had complex estate issues; 26 heirs of multiple generations potentially requiring multiple estates to be opened just to get permission to make a claim.

It doesn't have to necessarily specify a dog.Something to consider, though, if the current landlord knew about the dog and didn't do anything, this could be considered a waiver of any kind of objection to the dog - meaning they gave permission to have it by not objecting to it.You might want to see about getting insurance that will cover the tenant with the pit bull before closing.  

They make up ideas that seem to give them permission to withhold rent for one reason or another.

(they put their own in and moved mine to storage with my permission).

PM's typically have a $250-500 threshold that allows them to spend without your permission. 

But, what I'd do is give them written permission to have a long term guest over the summer.  

One thing I would insist on, is that no money - at all - is spent without your permission.  

And yes, apparently they changed the deposit amount on the leases starting with the 3rd lease to match what they had collected, but at no time discussed with me what was happening or sought my permission. 

If that doesn't work, you might ask him for permission to install one at your own expense if it's that big of a concern.