"in das" also means "into". "ins" is simply a contraction of "in das" in the same way that "it's" is a contraction of "it is".
Personally, I'd have translated "Sie passt nicht in das Auto" to "She does not fit into the car" rather than "She does not fit in the car" - but you'll hear either and they're both meant to mean the same.
No that wouldn't work. "nicht" has to come after the verb in this case (and many other cases). I actually find it pretty hard to explain when and why "nicht" is placed there, and while trying to find a good website answering exactly this question, I found one where the information given isn't even correct.
However, this one should be helpful for you (or anyone else with the same difficulties): http://www.deutschseite.de/grammatik/negation/negation.html - Scroll down to the section titled "The position of "nicht" in a sentence".
Yeah, it can be sometimes a bit arbitrary. In this case the "fitting" is an action, where you are trying to shove a fat person thru the car door ;) Therefore -> accusative.
Passen is a bit tricky. If you look it up in almost any grammar book it says that passen is always accompanied by dative ("Das passt mir gut"), but that is only in the one meaning of the word ("it suits me" etc.). If you would for example talk about a person fitting a sports team: "Er passt nicht ins Team" would be the right way to say it and "im" would be absolutely wrong
In this case a more literal translation would be "She does not fit into the car", meaning that she cannot get into it. "Sie passt nicht im Auto" is also correct, but would mean that, even if she can get through the door, there isn't enough room inside. In the first case, she's outside of the car and you're contemplating her moving into it; in the second, it doesn't matter where she is, you're comparing her size and shape with the free space in the car without considering how she gets there. This would be a relevant distinction if, for instance, she could fit inside the car but the doors are welded shut and she won't fit through the windows.
Shouldn't what be in the genitive?
What part of the sentence would you have expected to be in the genitive case, and why?
sie is in the nominative case, since it's the subject of the verb passen.
Auto is in the accusative case after in (she does not fit "into" the car --> destination).