They will ALWAYS ALWAYS ALWAYS end up wanting a favor of some kind that puts you in an awkward position, and they will expect special treatment because they are your pal. 

Most states allow tenants to leave any day of the month as long as they give notice 30 days prior to vacating, and the landlord is required to prorate rent through the day of departure.Your tenants failed to pay any prorated rent for the month of January so I would not give them any special treatment.

It will ask the medical professional if the tenant:1) has been under their regular care for treatment of their disability;2) that the tenant meets the HUD definition of disabled;3) that they require a companion animal in order to be able to live comfortable in your unit, because of their disability.You have every right to require them to meet the definition of disabled and to have a bonafide medical professional verify that they are disabled and require a companion animal because they are disabled - in order to be able to live comfortably in your unit.If the go to HUD and complain that that big meanie landlord didn't let me have my two "service" dogs, HUD will not immediately believe them.  

So you can't charge additional for an ESA, but surely you should be able to have an addendum, OK call it an animal addendum to require general things like the tenant is still responsible for cost of any animal damages, or flea/tick treatment of the property when they vacate, or keeping up with vaccination, must pick up after the dog etc...on the other hand, let's use "pick up after dogs" as an example...if the tenant is blind and has a seeing eye dog, how would the tenant go pick up after the dog? 

The place has been treated, preventatively, on a regular basis since the heat treatment years ago. 

It's truly an eye sore!

I would not care so much about scratches, but I would not want the puppy to go potty on the floor.Are their special floor treatments which I can request them to perform upon the move out?

You are being too lenient and you are also showing preferential treatment, all tenants must be treated equally.

I recently had a tenant that decided they want to break their lease and move to another city for treatment.

You can tell the OP he got the best possible treatment in the best of all worlds all day long, doesn't change that.