Translation:Vera Ivanovna is our favorite teacher.
Учитель is masculine. You can also say "наша любимая учительница", but it is common to use masculine professions for women.
I asked my Russian girlfriend. She says that when you approach a female teacher, you would address her using the feminine word for teacher. But when you talk about her with someone else, you'd usually use the general masculine word for teacher. (I hope I delivered her message right)
You never say anything like "hello teacher" in Russian. It is always a name with patronymic.
Maybe it was a bad example, so I deleted it. But i don't understand what your last sentence meant.
If you have a teacher called Марина Сергеевна, you'll always say "Здравствуйте, Марина Сергеевна!" and never "Привет, учитель(ница)". Looks like you misunderstood your girlfriend. No one uses the word "учитель" or "учительница" to address a teacher directly.
Pretty much. It's rather interesting how in Russian culture teachers are not addressed to by their last names, but the students are.
I first learned that these soft ended '-тель' forms take the gender of the person that they refer to.
It's was seen as necessary part of the 100 or so years of egalitarian socialism that most recently sculped the language-- aided by the fact that almoat all professions are of this gramatically genderless '-тел' form.
I was at МГУ, so I think this is at least 'officially' authoritative.