Wife Attending School in a Different State

My wife and I both are grad students living in the USA as F1 visa holders, but we fall under the category of resident aliens for tax purposes. We both were attending grad schools at DC.

 

Unexpectedly, wife's professor decided to move his whole lab to a university in Michigan. Since she is in her final year, she is still considered a student at the university in DC, and she is officially an intern at the Michigan university. Because of this, she still gets paid from the original university in DC (that is how the grant funding worked out for her).

 

She now rents an apartment at Michigan, and I am currently renting an apartment in DC.

 

1. Should we file taxes next year as married filing jointly? or married filing separately?

2. Since she is still getting paid by DC (her W2 is going to come from the DC university), is she considered a person working in DC even though she lives in DC?

3. Since we are still F1 visa holders and our primary intention in the USA is to attend school, how we determine where our "tax home" is?

4. May be this is not relevant here, but should she change her state ID to Michigan? She does not plan to own a car during ~1 year stay in Michigan. Her current license has my current DC address. Neither of us own any vehicles and nor plan to own one before we graduate.