Why and how to go to the US for a PhD-exchange semester? Context: My wife and I are both PhD candidates in good-but-not-major eastern European universities, aiming to defend our theses in fall 2017 (i.e., in 15 months). She works in accounting, whereas my research focuses on engineering science. She has been accepted in an east-coast US university for an exchange semester as of January 2017.
Since for obvious personal reasons I do not want to stay one ocean away from her during 6 months, I'd like to visit US-universities at the same time (+/- 2 months).

Questions:
Exchange semesters are not common in my lab. So I lack perspective on how and why to go to the US. So here are my questions:


What is the purpose of visiting another university at the end of your PhD? What would be expected of me if I do go there?
Must such exchange be formally validated/accepted, or is a casual agreement with the lab PI ok? (N.B.: I am financially independent)
Can I directly contact "the one" Professor (of international and cross-fields renown) my research and saying that I'd like to collaborate, or is it mandatory to be recommended? Are there tacit how-tos? 
Should I mention my wife's situation?


Note that I would have wanted such an exchange, even if my wife would not have been accepted for this exchange. However, her situation imposes the period.
My scientific supervisors do not have contacts in the US.
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.