Well, yes and no. Classical Latin doesn't have a locative case per se. The original suffixes of the locative case were transported from the Old Latin and then merged with the genitive. Therefore it's actually a genitive in the function of locative. It's worth mentioning that this form is used just for the names of cities and some islands and for some frequently used terms (like "domi"). You can't decline other words in this way.
Yes, it is. Roma, -ae, it is the locative form of Roma.
Roma Romae Romae Romam Romā Romae
Yes: that is a rare form of "locative" (an ancient case that has the same terminations of genitive case)
While Romae is the genitive spelling of Rome, Romae is not in the genitive case here because it is showing location.
It's "habitas" = 2nd person of the singular (you, "tu").
-ne is only a suffix, as in lavarse in Spanish, the "se" is not a part of the verb, the ending is still "ar".
Why "Don't you live in Rome?" or "You live in Rome, no?" wouldn't be correct?
Differences in the meaning between the English "Don't you live in Rome?" (I'm certain you do live in Rome, or you should live in Rome in some other contexts), and "Do you live in Rome?" = neutral way to say it.
It would really be great if the cases were written under the words' translations. It would help to know what we are dealing with and facilitate learning.
The pronunciation is wrong. AE, in Latin, sounds like E. It's a vocalic ditongue. It should sound like "RomE"
It sounds like this course is following classical (original Roman) pronunciation, in which case the pronunciation "Roam-eye" is correct. I've only heard "Roam-eh" (like the name of the letter A) for post-classical dialects. What dialect uses "Roam-ee"?