First of all, I would NEVER allow my children to be on the social networks in the environment we live in today. I think many parents are ignorant of the dangers their children/teens are facing by having a social networking page. Once my children are teens I would consider allowing them on, but I would have to be able to monitor ALL their activity and who their friends were and who they showed pictures to etc... Its not overreacting, its PARENTING.
That said, I am friends with lots of teens that I know from church or when I was a TA. I am friends with them in real life, trying to mentor them and be there for them, so this is just another outlet for me. I have a personal burden for them (similar to your nieces) and wish to make an impact.
You certainly have the right to "friend" whoever you want to. My rule is that we have to be friends in real life, so I don't accept people who I don't really KNOW.
As far as student teacher relationships it certainly can be inappropriate. Probably it would be best to have a secondary page to interact with them on, and not include them into your general everyday posts, jokes, pictures, etc... Many people forget who all is looking at their page and don't keep the content appropriate. THAT is the key issue, not if you're friends but what things you're sharing with your underage friends.
Our society is definitely TOO casual. My husband is a physician and HATES the title of DOCTOR. As the youth leader at our church he allowed the teens to call him by his first name, something I would not have done, and so they assumed that they could call me by mine. When I was the TA at our church's school that all had to change and one girl in particular struggled with that. I did insist that she continue to call me MRS. but the inconsistency was confusing. As adults it is important to connect with the younger generations, children need good mentors that are not family, but we don't have to lower to their level to make an impact. We are not their friends - on the level that a peer would be - but rather role models, and we need to be intentional about that.