Should teachers call their students honey or baby or words like that ?

It seems wrong to me, but I think you should let her know.

Sounds like she is showing concern for you and your well being. It's okay for a counselor to say that to help build a positive relationship with you. As for a teacher saying that, unless your in kindergarten or Pre-K. Not so much. If your not comfortable with it you can let your counselor know.

Not really. But if it bothers you then ask her not to do it.

If she's counselling you rather than teaching you it's not totally inappropriate. But in the classroom it wouldn't be acceptable, except for very young children.

I think she's using it to show she cares about you, -  it's just the way of some women. I think they're both a little condescending, especially "baby." If you don't like it let her know. 

No. Sounds unprofessional and a bit too familiar.