Should teachers give advice?

No. They should teach their course subject to their students. Any advice given would be limited to the scope of the course.

They aren’t trained to be academic advisors or career counselors. They’re trained to teach. They care about their students learning the course they’re teaching. As such, they care about their students.

See a different school professional for other concerns.

Obviously they should give advice about how to do well in class, and about possible future study or jobs.

But they should not normally hand out advice on personal problems.

No. That's not their job. They're not trained to do that. Not being trained to do that, they are just as likely to do more harm than good. Getting involved in student's personal affairs by offering personal advice crosses the line of not just what is professional behavior but actually what is ethical behavior for teachers. Teachers are often fired, especially in post-secondary education, for crossing that ethical boundary and failing to maintain an arms-length professional relationship with their students--and rightfully so. Their job is to teach the curriculum, not to be counselors, not to become involved in personal affairs that subject them to not only giving out advice they aren't licensed to give out but also subject them to bias or the appearance of bias in grading and making the same requirements of all students (e.g., requiring all students submit the same homework and not letting homework slide for some students because of sob stories they've told that the teacher has allowed to corrupt their objectivity, professionalism, and mandate to treat all students equally).