He can do anything he wants ( as long as there are no laws dictating his billing).   

We see the risk in losing our equity, or losing the use of our equity, by having it stagnate in a house.Some of feel that by leveraging to the max is more risky to them, because they don't have the control/influence needed to reduce the risk.Some feel that taking on someone elses problem (NPN) isn't risky since they are able to have a series of controls in place to either sell, repair, or rewrite the non-performance.Still others have more control over flipping since their market and their rehab abilities dictate more control there.It isn't the direction you take to invest that's risky.  

You should take this time to learn your local landlord/tenant laws, as they govern your business and will dictate much of what you do.

Tenant's failure to vacate will be considered a Holdover Tenancy and subject to the Holdover Tenancy terms dictated in this lease.Depending on your location, this may or may not be enforceable. 

Introduce yourself as the new owner, say you want to assure them that you are going to honor their leases at the current rate but that the cost of financing and maintaining the building dictates rents be raised to market rates at the end of their leases.

The cost in flooring damage alone from their animal dictates it.Although, it sounds like it's resolved now - good job!

So, you need to see what laws dictate what you can do.If your lease says you can evict if they don't pay their fees - or the fees are written in such a way that it's part of the lease agreement, and then you can call it a breach of the agreement - then yes, you can evict for non-payment of fees.So, step 1 is to figure out what laws govern your situation.  

Markets dictate pricing. 

Assuming the rental is in seattle as OP is,   Seattle has an (annoying but its the law)  ordinance https://www.seattle.gov/civilrights/civil-rights/fair-housing/first-in-time dictating how rental applications must be processed -  This is the first-in-time ordinance.   

The market dictates pricing so simply check to see what comparable properties to yours are currently renting at and then notify your tenants that the rent will be $x beginning on (date) and a new lease will be needed.