"a rural area in Beijing" vs. "the rural area in Beijing"

"A rural area" implies there are many rural areas, and possibly separated from each other.

"The rural area" implies there is one rural area, or that all of the rural areas are so close they can be confused as one single rural area.

I believe both are correct.

If you said to me "a rural area", I would want to ask "which rural area?" because it sounds like you would be able to tell me which unique area you are from. However if you said to me "the rural area", I would assume you are also saying "all rural areas are the same and there is nothing unique about any of them" so I would not ask which rural area you are from.

Yes, your usage is correct.

"A"/"An" are indefinite articles, which are used when the subject of the article is not specifically identifiable by the reader/listener.

My question was answered by a StackExchange user (as there are many StackExchange users)

"The" is the definite article, which is used the subject of the article is specifically identifiable by the reader/listener.

My question was answered by the ELL site on StackExchanges (as there is only one StackExchange user with that username; thank you to mcalex for the example)

Note that the sentence does not need to identify the article's subject itself. "The" can be also used because the subject is unique ("the year 2020 AD"), or the subject can be identified by a prior sentence.