This looks interesting. I'll do that, partly to prove I'm still here...

1. The easiest one to answer in this case - Reigenleif would likely do a little bit of research into the item first, determine its powers, history and generally sate her curiosity as much as possible without damaging it... Then depending on what she finds, she would either give it back and be happy with whatever reward they give, or if it seemed particularly interesting, she might even pay them, possibly with gold, possibly with something she determined equal value, for the privilege of studying it... If it really is that powerful, its highly unlikely she would give it to them without a comprehensive study - which may very well prove they are not the rightful owners, at which point she'd bluntly inform them of their being mistaken. Once she had studied it she would probably hold onto it until she could file it away in the college of Winterhold's collection.

2. A hard one for Reigenleif, she would have a lot of trouble hurting kids, no matter how fanatical. She would most likely use defensive magic to hold them off and either retreat to come up with a new approach or (maybe even abandoning it if not particularly invested in the mission) or if she really believed in the mission, attempting to rush past them and risk being flanked. She is fairly confidant in her ability to hold off unskilled opponents with a few well placed spells.

3. Basically a constant of living in Nord cities... She would most likely ignore him and keep going. she might even have an alteration spell specifically to help clean her clothes come to think of it. If she herself were drunk, and the hangers on were particularly rude, it might be a different story. She is not above a sudden, flashy display of magic to humble random drunks, hopefully without causing too much of an incident... She has been known to loose a little control of the precise strength of her spells when drunk after all...

4. She would likely have given the poor man gold earlier - gold isn't a very valuable thing for her, she only keeps what she needs to buy supplies generally. Truth is fairly important to her however, and I think it would have to be fairly extreme irritation before she would lie to get the man arrested, even if a guard asked her. Not to mention she generally feels very out of place in cities, and keeps to herself quite a bit when there, so she may very well pretend the guard had not said anything to her. If she was feeling bold that day she might tell him to find someone else to do his dirty work for him before moving on quickly.