Should I be fluent in the languages of the countries in which I wish to teach abroad?

It depends - will you be teaching english? I taught english as a second language in South Korea, and didn't know a word of Korean when I started. The students spent half of class with a korean teach (who taught english) and then with me (who of course taught english). At times it could be a little silly when they needed clarification or couldn't understand, but not being able to speak in korean with them actually sped up the learning process. My students couldn't fall back on their native language. That was only my experience though - ask the head of the school or program what they suggest, as it likely varies by the type of school and country. I'll also say that living in korea but not knowing korean often made the day-to-day challenging for me (lots of gesturing!) but that was what made it fun. It's valuable to be the odd man out and figure out how to make the most of the experience.

As an English teacher it is not required to know the language of your students. Most schools even prefer you dont speak in the native languages as student should be practicing English while they are with you. Personally it does help to know the basics of the language. It makes buying food, using public transport, etx much easier for you personally.

It doesn't hurt, but it's by no means strictly necessary - especially if you end up in a big city.

On the other hand, learning a language when you're immersed in it is a lot easier (arguably it's the only way to really learn the language anyways). So, it'll be easier to pick up once you're there.