1) I agree with Paul that people near me tend to call our local mall by its name "the xxx Centre" and wouldn't refer to it as a mall. But I think the word mall is commonly understood and wouldn't be wrong, especially if you were talking about one of the really big centres in London.
2) I would disagree with Paul because we do call it the DIY store. To me, a 'shop' implies a smaller establishment, whereas when referring to one of the bigger superstores like B&Q, we would call it the DIY store. Ironmonger is a very old-fashioned word, I think there are only a few small local stores that would describe themselves that way, and they tend to mainly sell metal objects such as fastenings and tools.
3) Agree with Paul that drinks cabinet is most common, but liquor cabinet would not be wrong. Drinks cupboard sounds wrong, no-one would say that.
4) agree with Paul, they sound like very specific technical descriptions of tread patterns for tyres which would not be in common usage. I just looked on a popular British tyre chain (Kwik-fit) and they don't have any description like that on their front pages.