"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.
Native English speakers often colloquially use "up" to mean "located north" and "down" to mean "located south". If my friend was going to school in the north I would say she was "up at school", if her school was south I might say "down at school". I might not even really know where the school is and still say "up" just because it makes the sentence flow better.
In this sentence it essentially has no meaning; you could understand it to mean exactly the same thing as:
"Does she seem happy at school?"
(You can say "over" to mean "located east" or "west".)
Best Answer
Usually, this would be worded as "highest school completed", but it is sometimes shorted to "highest school". Generally, the answer would be the level, e.g. if you have a Masters degree but not a Ph.D you would put down "Master's Degree".
Unless there are spaces for further details like degree attained, field of study, years attended, or GPA. Then I would assume that it is asking for the specific institution.