Even if you decide to not act on anything you find in order to keep your "good tenant", you should at the very least be aware of the skeletons, if any, in the closet of this new tenant.

Go through a crazy biker tattooed boyfriend stage, with motorcycles parked on the lawns and a grow room in the closet?

Savvy tenants aren't going to just believe you when you say it will be cleaned and painted and in pristine condition and available on the day it's supposed to be - when they are stepping over someone dirty laundry.And, they won't feel comfortable looking inside closets or the garage, etc.And, there's the potential liability of your current tenant claiming some expensive jewelry disappeared during your walk-throughs.I know the idea of having to eat a month's rent in-between tenants hurts, but I truly believe you get better tenants, who will not be as likely to cost you in problems, if you clear out the last one, get the unit in beautiful show condition, then start showing it.  

And by not personally meeting them at an appointed time, and asking them why they're looking for a place to rent, and watching if they notice or care about how big the closets are, or what the parking situation is, etc., you could miss some big clues and red flags.  

Does the 4th bedroom have a closet, if not it is considered non-conforming and may not add as much value as a conforming bedroom.

Do they look to see how big the closets are? 

I know that my rentals CAN accommodate more people if someone sleeps on the sofa(s), the floor, in the basement, and in a large closet someone could use as a bedroom, but I reserve my rights to ask who will be living in the home.

In closet?

The former kitchen opened to a giant empty room in the back, that you accessed by walking into a large closet??

I don't care for portable space heating units in a rental so it would help if you could elaborate on what you mean by " not requiring them to buy a room heating unit'.