Do OTHER colleges know if you've applied Early Decision or Early Action elsewhere?
Question: Do all the other schools know what your early decision/early action school was? Does this play a role in whether to admit you or not?
No, in theory, admission officials will not know if you applied via Early Decision or Early Action to another college. I say "in theory" because sometimes high school guidance offices put information on transcripts that can reveal a student's application history. The transcript may not say specifically, "Applied ED to Dartmouth" but there may be a notation that a transcript was sent in October to Dartmouth and then to other colleges in January, which suggests that this student was an ED candidate at Dartmouth who was either rejected or deferred.
Of course, that doesn't happen very often, so colleges really shouldn't know where else you may have already applied.
There is a group of highly competitive colleges that do share their admitted Early Decision applicant list with each other. But this happens only after the students have been accepted and have agreed to enroll. Thus, if you were deferred or denied by one of these schools ... or if you were admitted but had to turn down the offer due to finances ... then your name will not be on that list.