Growing up in Philadelphia, we frequently used the phrase "go into town" to mean going into the busy center of the city, the business district. This usage may just be regional, but in general I'd say that the context makes a big difference on whether "city" & "town" are exchangeable or not in American English.
Yes, it is incorrect. "Дома" and "домой" are unusual in that they are used as adverbs - just like English "home" in sentences like "staying home" and "going home". You should not generalise these examples to other nouns. By the way, German also has those expressions: "zu Hause" and "nach Hause", which don't follow the usual pattern.
Один город. "Я живу в одном городе" is wrong, nobody says this way. This sentence from the task has another meaning "I live in a city and not in a village" or "I am a city dweller". This is just an abstract sentence. Here it is not referring to a specific city. So It doesn't need the word "the" here.
But you can use this word to say that you live in this city. In Russian it would be "Я живу в этом городе"
Both "a city" and "the city" could work here depending on a context. "I live in a city" is something one may say to indicate that he/she does not live in the countryside. No specific city is implied here.
"I live in the city" is something a New Yorker can tell a person from the Long Island suburbia. (This is not specific to NY, of course, it simply means that all parties in the conversation understand which city it is).
In Russian, you would use exactly the same sentence in both cases, with the context taking care of the precise meaning.