Lo and behold our house's value is identical to what it was last year even though our recent valuation (also done as a refi through Chris a few months back) doesn't match that number...like he said those numbers don't matter.

A full gut in Port Richmond or West Philly is not going to cost the same as a full gut in South Philly even with an identical sqft. house, and a rental rehab will cost significantly less than a flip rehab.

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.

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.

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. 

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. 

In this case if each unit was identical then the rental portion would be $200K and a 1031 into that property would be fine.The down payment for the entire purchase can come from the 1031 proceeds.