A good chunk of a teacher's job is to build appropriate relationships with your students. Students don't want to learn from someone they dislike, and you have significantly better learning outcomes when the students feel safe, accepted, and cared about. Appropriate nicknames, like Tim for Timothy, help in that relationship building. I don't know what your position is at that school, but Wisconsin teachers are literally taught stuff like this in college so that we know how to manage a classroom with the best learning outcomes and the fewest number of behavioral disruptions. We are taught how to keep those relationships appropriate and healthy, although much of that is just common sense.
Yes, you should separate work and home life for both your own sanity and for modeling good boundaries and work-life balance. But that doesn't mean you have to drop your decency at the door. At the end of the day, the goal is learning, and not being a douche is one of the easiest ways to get to that goal.
Extracurricular activities are an extension of these same principles, not an exception or something with a different set of standards. I think you might be mixing up appropriate relationship building with inappropriate fraternizing, and I'm concerned that you are having difficulty finding that line.
I definitely agree that there aren't enough resources given to teachers, but the expectation of using common decency to reach the goal of educating our students is not too high of an expectation. Focus on the end goal. How you get there can vary (assuming it's appropriate), but you are still trying to reach the goal of educating the students. If your teaching style is prohibiting people from reaching that goal, why wouldn't you change it?
It's nice to think that as an English teacher, I only have to worry about how well they can interpret the modern applications of the lessons in Macbeth, or whatever literature we're studying, but in reality, teachers are teaching a whole heck of a lot more than their specific subject area. We're simultaneously modeling how to behave appropriately, teaching how to navigate complex social situations, and mentoring students on how to achieve their goals and deal with set backs. Teachers have always worn more than one hat. It's not only an expectation for the job; it's an absolute requirement for success.
Should they earn more money for having to do all of that? YES! That's why we've been complaining about the low pay and lack of resources for at least 40 years. The effort and skills are non-negotiable. Kids shouldn't get a crappy education just because some politicians are using their teachers' wages as political leverage. People go into education knowing that the pay sucks, but they actually care about other people and future generations. They don't go into just for the paycheck, and I don't know a single educator who wouldn't put in some extra effort to help a student succeed.
You're basing a lot of your opinion on the assumption that kids come to school ready to learn and healthy. The reality is that parents and home lives come in a wide variety of flavors. Some parents do exactly what you said: dump on teachers with their own expectations on how students should be handled. But others don't get involved at all. Some don't care about their child's life beyond how it affects them. Some are so busy working to make ends meet that they don't have time to be much more than an absent parent. No matter what life the student has, it's still my job to give them a quality education, so of that means giving them a granola bar or calling Joe Suzie, then that's what it takes.
We're basically fighting for the same thing here: better pay, better resources, and support for teachers so that students can get a better education. The difference is that I don't think students should get the short end of the stick for something they can't change (i.e. low pay), whereas you'd rather a teacher not do extra because they aren't getting paid to do extra. But my method reaches the end goal of educating students well, and yours instead basically says, "Reach the goal or don't. I don't really care since I did my part."