Translation:Many important people are walking at the American department stores.
As an American that visits department stores, this sentence makes me feel warm and fuzzy.
Okay, let's go over the differences:
Mellett is a postposition which is usually translated as "beside" or "next to". The "side" part in "beside" is the important thing here: you are at the side of the object, not in front nor behind, above or below. If you say "A színház mellett lakom", you're saying that the theatre is your neighbouring building.
-nál/-nél is a suffix and is usually translated as "at" or "by". -nál doesn't care exactly where you are, just that you are in the immediate surrounding of the object. If you say "A színháznál lakom", I know you're living close to the theatre, probably in the same street. The suffix can also be used in that less literal meaning of "working at" a place (like in English, hey!), but other than that it's used mostly in the spatial sense.
Strictly speaking, "working at" some company doesn't necessarily mean that you work inside of the building. You can work "at the bank" (a banknál), but your workplace might not be "in the bank" (a bankban), maybe because you work in customer service, where you can stay at home and receive calls there.
When does one use many and when does one use "a lot"? Usage seems inconsistent.
You usually use "many" with countable nouns and can use "a lot" with either countable or uncountable nouns. The exact rules are weird and complicated, and I'd just suggest to report the missing expressions here. The course is still young and misses a lot of (many) possible answers.