Why 'wolf down' but 'eat up'? ;-P Don't expect such collocations to make sense in any language, since they usually are a convention that native speakers learn from their parents and not down to the logic of the language – In Polish we have "Uczy się w szkole", "Uczy się w liceum" but "Uczy się na uniwersytecie", even though there is nothing inherently wrong with "Uczy się w uniwerytecie"; we just don't use it.

Oh, and don't also believe in folk etymology that this have anything to do with the fact it is tertiary education(which I seen on the Internet) – we still say "Uczy się w szkole policyjnej"(a tertiary education establishment) and "Uczy się w Akademii" returns 16,800 results in Google vs. 18,700 for "Uczy się na Akademii", so they exist almost equally – as I said, no logic to it. ;-)

As per what Emwue said, I'm not going to say this is necessarily logic you can follow consistently, but when choosing between 'do' and 'na' for the English meaning 'to', you may have heard that 'na' usually goes with larger places (na lotnisko) and 'do' often with smaller places (do domu). In this case, it's not hard to imagine a university as a very large place, so use 'na'. Again, I don't say this to imply this logic holds widely for choosing between 'w' and 'na' for the English meaning 'at', but in this one case, it might help you remember?

In the PL->ENG version of this sentence, you cannot know that. Both he and she are accepted, you just choose one.

But also "na poczcie" (at the post office) and "na stacji"/"na dworcu" (at the station, despite the fact that it may be closed space, at least "dworzec" because "stacja" often isn't... "na lotnisku" (at the airport) although that's also a building... the general rule you stated is right, but the number of exceptions isn't that small.

"studiowa" is not a word.

"studiować" is the infinitive form, of which "studiuje" is the right form here.