I would say training at the language center. Also, you need an article before "language center" - either "training at a language center" or "training at the language center." Which one you use depends on the context.

I should add "tutor" usually refers to a person teaching one student at a time (or at most a very small number). So, a tutor could certainly work at a language school, but when he/she is "tutoring" at the school, it's probably only one student at a time.

1) either is acceptable -- my sense is "at" is more common. 2) instructor, teacher, trainer, tutor, professor -- these are all words that might be used. It largely depends on what titles the school uses.