What is the best way to verify their identity and run a background check? 

If you wouldn’t buy the identical house for a rental don’t keep the one you already own.

Not one theft in over 12 months!

I have a person who qualifies on paper that I gave a showing. They asked me to email them an application, which I did. It became more clear this person may be mentally unstable. They expect an immediate response but I...

As you can see by the interior photo, the house is in great condition, with the rest of the property in identical shape.

You will need to remove those for winter and put them out in the spring.I know people mention liability, but I would also be concerned with maintenance, damage and theft.

Also for various reasons our properties would not qualify du to requirements of being rented for a period of time or owned in a particular company and the requirement that the replacement property and the original property have to be identical owners. 

Major lessons learned:1) Sometimes the biggest disasters lead to the biggest upside.  2) I filed two insurance claims here - one for the property damage and one for the theft of electrical & plumbing. 

Downside is I can't verify their identity but I like that I know their last name upon inquiry.

One thing to keep in mind if you look at an FHA based loan such as as 203k, you may have an Identity of Interest conflict.