The 哪 "na" and the 儿 "er" make a single sound when put together: 哪儿 "nar"

if it's a component of a word like 哪 it doesn't exactly become silent but rather reduced to a sound similar to an english "r" in a word like "room"

It is merged with the second to last character to form the sound "na-er."

Because the verb live doesn't have a progressive form (at least it wouldn't make sense to me), you either do live there or don't you can't live somewhere for 10 minutes and you don't live there anymore. (I hope I am not telling any nonsense xD)

Almost. Just take away "at". They live in a place/city. They live at an address.

LOL, "Where do they live at?" is bad English. Should not end a sentence with a preposition.

I think "ta men zhu na li" or "ta men zhu nar" are both accepted. At least in Shanghai as long as I know you don't need to use "zai"

I choose "where they live" and idk why is wrong! I think they translate how they prefer