Education

For your scholarships - yes, it is accurate that a foreign student can include this type of income as foreign earned income, thus making it eligible for the Foreign Earned Income Exclusion (FEIE).  You are not required to pay taxes on income that is excluded through the FEIE, however you must not be an exempt person who isn't subject to any taxes in the country in which you reside.  

For your wages from your part-time work, you have properly listed this as foreign income, to be excluded under the Foreign Earned Income Exclusion (FEIE).

Since you can only list one employer on the FEIE, listing the University sounds reasonable as your primary employer.

When taxpayers abroad have more than one employer, it is proper to list the primary employer or the most recent employer (in the case of a change in employers.) 

While the questions about your living situation follow the question about the housing exclusion portion, they are actually related to the questions on the form that qualify you as a bona fide resident.   You can exclude your income when you are either physically present in another country (or countries) for at least 330 of 365 days or when you are considered a bona fide resident. 

The answers to the questions about where you live and with whom are part of the determination for whether or not you are a bona fide resident.   The idea behind some of these questions can help you better understand how to answer them in most cases.  What the IRS is looking for here is evidence that you are living indefinitely in the other country rather than for a specified temporary period of time.   Indefinite does not mean forever here or that you don't intend to return to the U.S., just that your not visiting for a specific limited time period.

If you qualify under the physical presence test, these questions are a little less important.  

Now, for the specific questions that you have about this:

• Did you have any family members living with you in your main home? In your case, since your U.S. based family is not with you I would answer no.  (You didn't bring family with you - but it may not have been appropriate to bring them.  This typically refers to the typical U.S. family unit of spouse/children.) 
• Describe your housing situation.  I would select a rented room, even though you are living with part of your U.K. family. This most accurately represents your situation at the present time.
• Does your visa limit the length of time you can work outside of the U.S.?  This refers to the length of time your visa or work permit is valid and whether or not it can be renewed or has any restrictions.   In your case, I would say that your work permit is valid as long as you are also a student and is valid for part-time work only, (if I understand correctly) and give the visa expiration date where asked.  

Thank you for your question.  Hopefully, you can wrap this up today so that your long week of tax work is over and you can return your energy to your Ph.D. studies and other activities!