It is correct.

Yes, it is needed.

It is a common mistake to mix up 哪儿 and 那儿. 哪儿 means Where while 那儿 means There. As here is a Yes-or-No question, 吗 is needed.

You're thinking of the eternally confusing 哪儿，rather than 那儿。note the extra box. I believe 哪儿 is third tone, while 那儿 is 4th

Without it the question becomes a statement.