What type of character (caring, trustworthiness, respect, fairness, and etc.) education should teachers implement in their school?
This is, of course, a matter of opinion. In my view, it is not the place of the education system to teach character. That is a job for families, churches, and other private institutions. There are, however, a couple of things that could be called character that teachers should help to instill. These are things that teachers should teach for the sake of education in general, not for the sake of character education.
For example, teachers should help to teach students to be honest when it comes to their work. They should help teach children not to cheat or plagiarize. This is because cheating robs the student of their chance to get their own education.
As another example, teachers should teach their students to have respect and compassion for their classmates. Again, though, this is necessary for general education. If your students do not respect one another, they will treat each other in ways that are very disruptive. It will make it harder for academic education to take place.
Thus, I would argue that teachers should only teach character when it is important for academic purposes. Outside of that, it should be up to parents to decide what sort of character education they want their children to have.