Why and how to go to the US for a PhD-exchange semester?
If you have only EU passport, you may be eligible to stay up to 90 days in US using ESTA in order to "attend short-term training (you may not be paid by any source in the U.S. with the exception of expenses incidental to your stay)". Otherwise appropriate visa is needed (e.g. J-1).
Contact the professor directly. You can't lose anything. 
The purpose could be to improve your thesis, initiate networking, start some collaboration etc.
AFAIK, I didn't need any formal agreement - only when I applied for grant, I submit a formal letter of the host Professor that he will welcome me.
Your wife situation is that you'll be there for your personal interest anyway and that's opportunity to collaborate.
Ask your university department for international relations, whether there is some agreement with the US university and yours, there might be opportunity to cover some expenses of your stay there. It's better to get at least partial funding from your home university and claim that your home university pays you during immigration control and makes it easy with the host Professor lab. 


I did internship after my PhD in US and stayed only 1 months and above mentioned points saved me from possible disappointment during border control. Host Professor welcomed the fact that I was independent financially. I used a small university grant from "eastern european" university which covers part of the expenses - the flight tickets, all other I paid from my pocket money.