The Times World Reputation Rankings consistently rank Stanford, Harvard, and MIT as the top 3 in the world. All 3 do well in “global prestige.”
Stanford offers a summer quarter, in which students can take classes and stay in dorms, as well as summer room and board without classes. I did this one year. My roommate for the summer was an international student, who was from a royal family in Africa. Dorms do officially close during winter break. Stanford offers special off-campus housing for students who are unable to find alternative housing during winter break.