Translation:The elderly police officer comes down to the shop.
Does the -hez here mean the policeman ends up outside the store, not in it? If so, would üzletbe mean the policeman ends up inside the store?
No, I don't think it implies any such thing. üzlethez merely means the officer is coming toward the location of the shop (from some other place), and lejön implies the officer completes coming to that location, but without any implication of going inside.
You wonder, where does the officer actually end up: the answer (of course) is az üzletnél, at the shop. Previously, the officer was definitely not near the location of the shop, but is now at the location of the shop.
-nál/-nél is often described as "by" or "near" rather than "at". But don't mistake this for a literal translation. "next to" or "beside" is definitely az üzlet mellett, not az üzletnél. Az üzletnél is translated more as "near the store" primarily because "at the store" implies (in English) "in the store" which is a meaning that -nél does not share: az üzletnél merely is referring to "at the location of the store" and not "at some other place"
So, az üzletnél is where the officer comes to. Not az üzletben or az üzlet mellett
How precise are the Hungarians in using le- and fel- when they move slightly up or down? For example, if you go from hilly Buda to Pest, or from Balatons northshore to its southshore, is that already lemenni (going down), or is the height difference too small for that?