a department store is a different type of building to a shopping centre. a shopping centre has distinct different shops with their own entrances and normally a central access hall area, whereas a department store has no walls or doors between the areas and all items from different producers can be paid for at one till. Maceys or bentalls or John Lewis are department stores. Bluewater and Lakeside and Westfield are shopping centres.
Uj relates to Finnish Uusi, both meaning "new". Every time I visit Hungarian lessons I am struck by the Finnish words I find, the familiarity, AND then the strangeness of the rest of it, like some weird Turkish or some other strange language far removed from both Finnish and other western European languages. Hungarian is both very familiar AND very strange at the same time, for me. Fun to learn!
The best translation of "áruház" would be "store". It usually refers to department stores, but "webshops" and for example "play store" are also called "webáruház" and "play áruház" respectively. I wouldn't use "mall" as a translation here. Malls in hungarian are usually called "pláza" or "bevásárló központ".