In response to how Tonks behaved in school,
I've actually been toying with the idea of writing a "Tonks era" school fic, seeing as how there are so little of them and so much creative room to work with, so to speak.
I imagine her as being (especially when she's younger) being known by the whole school for a few things.
1) quite obviously, she's a metamorphagus which we haven't encountered anywhere else in the whole book series, therefore she's rare in that way already.
2) What I think people forget is that she went to school after Voldemort's downfall and all those leftover Death Eaters were being caught. Also, we know that Sirius considered Andromeda his favorite cousin and it can be assumed that he might have visited at some point which may or may not have been known to the wizarding community. Add that to the fact that she's related to around half of the Death Eaters, it's likely she would have been seen as "the metamorphagus who's Bellatrix Lestrange's niece and Sirius Black's cousin"
If this is true, which there's a good chance it is, she must have felt compelled to prove herself and show where her loyalties lay which is where I think she got the idea to be an auror during school.
That is just my take on it though :]