"Educated at university" isn't a special expression; at university is simply a spatial modifier showing where the education happened.
It's the same as saying something like "made in China" or "bought at the mall".
(a) [does it mean they] necessarily graduated from/received a degree at the named university,
It implies it, unless further information has been given to suggest they didn't. He was educated at Oxford is taken to mean He has a degree from Oxford University. I suspect the assumption is that if you didn't complete your education or didn't pass your final exams, you don't count as being "educated" there.
Note what I said about "unless further information has been given to suggest they didn't". Context is everything, and you could of course say something like, He was educated at Oxford for the best part of a year before the stress got to him and he returned to driving taxis.
Like with a lot of other constructions, you can also deliberately mislead people: He was educated at Oxford could mean that he didn't get his degree, or that he got it from Oxford Brookes University (rather than The University of Oxford, which is the big prestigious one), or even just that he went to secondary school there.
The deception arises from the fact that you would be using the literal definition and the fact that the literal definition is not how the phrase is normally understood.
(b) can it only imply the person graduated from the named university,
(c) can it mean that the person studied there and left (either to another uniiversity or simply dropped out), without having graduated,
Answered these in part (a).
(d) can it mean a current "undergraduate" or "graduate" student (or their BrE equivalents) at said university?
It would help if you provided a full sentence as your example, rather than just a phrase. If you say he was educated at UniversityName, you mean that his education there has finished.
You could theoretically say he is being educated at UniversityName to suggest an undergraduate. However, I'm not sure I've ever seen this phrase used in this way; it would tend to suggest something along the lines of corrective education (because of the passive tense - compare it to he is learning Subject at UniversityName) and so you'd be more likely to see it referring to primary or secondary education.
Postgraduate study tends to be even more self-taught so in you would be even less likely to see the passive phrase being used in that way. In my opinion saying "he was educated" to refer to postgrad study would be extremely unusual if not outright wrong.
Best Answer
The chief pedagogical and administrative official in a U.S. K-12 school is indeed known as the principal, and this is the default term as well as the federal classification of the job.
Indeed it is, in many American dictionaries, the principal noun definition of principal. For example, AHD:
MW uses the broader sense in its full definition, but opens the summary definition with
Principal as both a title and occupation is almost universal in traditional K-12 state schools (i.e. public school), and no matter the official title of the official, this is the term by which the position will commonly be known. The title is not at all common in American postsecondary education.
There is considerable variation under other public and private educational models (e.g. charter schools, Catholic and other parochial schools, independent schools), and the principal may be known by a variety of other titles.
More than a few such schools adopted British terminology, regardless of whether the school itself follows any part of the British educational model; titles like headmaster / headmistress and head of school are uncommon but not unusual. But these are not interchangeable with principal, due to their connotations; an American is likely to assume a school with a headmaster is an expensive private school, possibly in New England and probably a boarding school.
In contrast, head teacher, which has become conventional in Britain, is relatively rare. A simple COCA search, excluding fiction, turns up the following counts:
(It is difficult to search simply on head and on principal, due to their multiple common meanings. But this makes the prevalence of school principal all the more striking, as the other terms are far less ambiguous).