Learn Hardcore Russian: Рядом с домом стоит закрытый магазин.

No. рядом on its own needs the preposition с plus instrumental to form the standard “next to” construction. Saying рядом домом would be ungrammatical.

What case is магазин in and why?

магазин is in the nominative case because it is the subject of the verb стоит (stands/is located).

Why is the verb стоит used here instead of something like находится?

Both стоит and находится can describe location, but стоит often implies a vertical or upright position (e.g. a building “stands”). находится is more neutral (“is located”) and would also be correct: “Рядом с домом находится закрытый магазин.”

What part of speech is закрытый, and what does it convey about the магазин?

закрытый is a past passive participle used adjectivally. Here it means “shut/closed” (not open for business), modifying магазин.

Could закрытый магазин ever mean “a private (members-only) store” rather than “a closed store”?

In most contexts закрытый магазин means “a store that is closed.” To say “private” or “exclusive,” Russian uses adjectives like частный (“private”) or закрытый по подписке (“closed by subscription”).

Why is the word order Рядом с домом стоит ... instead of Стоит рядом с домом ...?

Russian word order is fairly flexible. Fronting Рядом с домом puts emphasis on location. Стоит рядом с домом is also correct but slightly less focused on “next to the house.”

Why are there no articles (a/the) in the sentence?

Russian does not have definite or indefinite articles like English. Context and word order carry the functions of “a” and “the.”