One advantage of a/the big city is that there are a lot of places to visit

Generally probably "a". If you're talking about a specific city then it would be "the". For example, if you live in the suburbs of a certain city, then you would use "a" for any city in general and "the" for the city you live near to.

You could use either! "A" big city would imply any big city and it doesn't really matter which, whereas "the" big city would either mean you're talking about a specific big city, or I would also use it to refer to a "the big city" as a concept. Personally I'd use "the" and be referring to "the big city" as a concept.

In that context: a. But if you were talking about a specific big city like Stockholm or Hong Kong or New York… Then you would use “the” to reference that city.

Grammatically and semantically speaking, "a" is technically correct. But "the big city" is an idiom unto itself, so if you use "the" here no one will bat an eye.

It depends if you’re talking about a specific city or not. a big city would mean any big city. The big city would be a specific city based on context

In this specific example they both work but for other nouns than "big city" you would only use "a" -- here "the" works just as well because "the big city" already exists as a general concept in the culture/everyone's imagination 

In this case, "a big city" is slightly better. Using "a" makes it more general, referring to big cities as a category. "The big city" could also work if you were thinking of a specific big city in mind or "the big city" as a general concept of urban life, but "a big city" is clearer for a general comparison between rural and urban life.

"A" may be more grammatically correct, but in casual speech I'd be more likely to say "the big city". You could say either and easily be understood!

You've picked a phrase by accident it seems, "the big city" is a phrase used to refer to urban life, "life -in the big city" for example. It also can be "in the city". In general English grammar, this being an exception since it's a turn of phrase, the is a specific thing, a is a general thing. The city indicates one specific city, "you should move in to the city", this means London is you live in London or Glasgow if you live in Glasgow. "A city" is for any random city, "you should live in a city" could be literally anywhere that is a "city"

In some contexts “the big city” could be used to refer to a specific large city that you are near or has been previously referenced in the conversation. “The” signifies a specific city. It’s generally an uncommon thing to say rather than just using the name of the city. “A big city” would be used to describe any large city. It may be used when someone asks you where you want to travel. You don’t know where specifically, but you do know you want to go to “a big city”.

you would only use "the" if you were discussing a specific (but unnamed) city. "One advantage of living in a big city, is that there are a lot of places to visit." works for general use. While discussing living in New Your City vs. New Jersey, you could say "One advantage of living in the big city, is that there are a lot of places to visit." and here it's implied that New York is THE big city in the area, but if you haven't specified a specific greater metro area, it's unclear.