I am also curious

It is not incorrect: "A nő fent van és az autó lent." is an accepted answer.

It's called ellipsis. Languages love omitting things that are unnecessary and already understood from previous context. You can do it in English too. But if you can't be bothered by it, feel free to put it in. :) It's just as correct.

Yes.

No, in this case 'alatt' would be incorrect, because, in contrast with the English 'below', 'alatt' is never used as an adverb. Hungarian adverbs that translate as 'below' are 'lent' or 'alul' and there is a more archaic word to express the same: 'alant'. 'Alatt' is always used as a postposition, that is, it is necessarily preceded by a noun phrase "az asztal alatt" - except for the personal forms of 'alatt', ie 'alattam', 'alattad', 'alatta' etc. ('below me', 'below you', 'below him' etc). In these personal forms of a postposition, the suffix -am, ad, a, etc indicates the personal pronoun it belongs to (én, te, ő)

Generally speaking, "a nő fent, az autó meg lent" as a stand-alone sentence would be a non-sentence (hence unacceptable), since this sentence type (Subject-Verb-Adverbial) requires the presence of the verb - either "van" or another verb such as "áll" (stand). In practice however, the best (most concise) reply to the question "Hol van a nő és az autó?" would be "A nő fent, az autó meg lent", since the verb could be understood from the context, due to ellipsis explained by Hatcher.

The hard and fast rule: within the S-V-A sentence type, where S takes an adverbial complement to express a position, location, or some other adverbial meaning, such as manner, the verb "van" or "vannak" - ie the third person of "lenni" (to be) - is NOT omitted. In this case the verb is necessarily present. However, in the sentence type Subject-Verb-Subject Complement, where "van/vannak" is a linking verb between a subject and its complement to express an attribute or identity, eg. "ő egy férfi" (he is a man) or "ő magas" (he is tall), the third person, and only the third: "van/vannak" is omitted. This is true in both cases, when the complement is a noun or adjective. An example to demonstrate the difference: "Ő férfi, én pedig nő vagyok" (= he is a man and I am a woman) rather than "Ő férfi van, én pedig nő vagyok".

I'm sorry, I am not great at sentence structure terms, so I will not be able to give it to you using fancy terms, however, I can give you a simple answer. In this case pedig is used more as "and". when its not being used as and, it is usually used as 'however', 'but', or 'even though'. for example, "nem szeretem az almat, pedig finom". Whereas hanem is used more as a rough equivalent of 'not' "nem A hanem B." If it helps, remember that 'hanem' is just 'ha' (if) and 'nem' (not). In this example hanem could only be used if you were potentially correcting someone. "a no nem fent van, hanem lent az autoban" -"the woman is not upstairs, but downstairs in the car". I'm sorry, I am not a linguist so I can not give you a very concrete answer, but I am a native speaker. I hope this helps. Good luck.

See ExDuoUser's excellent reply to Mary above. :)