I would think it would be very easy to cross lines between professionalism and friendship, and accidentally share things that ought not to be shared with students. It's probably safer for teachers if they don't do it.
It depends. As someone who has been in the classroom I would say this is something you have to carefully consider. Firstly, what is the school policy? If you can you should consider customizing your setting so that the students do not see your posts that are of a very personal nature. Just type "teacher fired" in google and the top result you will see is "for Facebook".
I see no reason for them not to. I read something a few weeks back about a teacher who prevented the suicide of one of his students because they were fb friends. I can see that Facebook has the power for good AND evil in this respect, but I think it's ok to assume good until something starts to smell bad.
Yes. I think teachers have the sense to remain private on Facebook. They could have two profiles- one as Ms Smith and a their private one as Jane Smith. Maybe there could be a school Facebook group to limit the danger and keep everything open. I think it is important to build a relationship with students and keep lines of communication open . It may prevent online bullying, unsafe internet use and suicide. Social media is now a fact of life. Children need to learn how to use it safely.
A private relationship between student and teacher on Facebook could be dangerous and lead to grooming by an unscrupulous character. Unfortunately many conscientious teachers are so overloaded with paperwork, I doubt they would have time for Facebook.