Is it for an institution only, as opposed to "centrum" for the central part of a city? Or I can use "center" for both?
I'm not completely sure but I think 'centrum' is like you said, a central part of a place, while 'centre' is more like a mall or shopping center.
'A shopping centre' is rejected?
A 'centre' meaning an institution/building is almost never used on it's own. I mean, there are youth centres, shopping centres, leisure centres, conference centres, but if I pointed to a building and said "That's a centre", that would be a strange and almost meaningless statement.