You should have done your rent research prior to listing and definitely prior to accepting any applications. 

See if your lender will accept a realtors opinion of value.

Please accept my friend request.

Learn as much as possible about them, THEN strategize.I agree, I not only research who has an interest in the property, current address for them but then I go to to social media and try to find them.

Got an accepted offer but I’m curious what thoughts are on having a rental with a natural wood fireplace or what would you do to ensure safe or no usage.

They have placed a stipulation on their listings, and I quote "Seller will not allow interior access until an accepted offer is in place."

The rent increase is on the date you stated and you will accept a 30 day notice if they desire to vacate the property prior to the rent increase.Plus you have no loss if they do leave, they are a maintenance drag.

Neither party really has the ability to fully control the agenda so both are going to need to accept some significant compromise positions, and where that will fall is at this point just a pure guessing game.

BTW, the prior landlord informing the tenant of the rent increase could easily be viewed as a renewal offer and depending on the response of the tenant, an accepted renewal offer. 

If your tenant is late in rent or not caring for the property, it is because you have created an environment where it is acceptable.