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. 

How in the world can they claim there is mold then dictate when and how it will be remediated while you provide alternative rooming... and now they want A-Z... consult an attorney and get out of this situation ASAP.

I've let tenants dictate terms too much and that stops now.I've had to learn my lessons the hard way in this business.