I would use "linguistics department/facility" because, if you change the order, it's "the department of linguistics", not "the department of linguistic".
"In conservatory" doesn't work at all. I can't say that it would be completely incorrect to say "in the conservatory" but that denotes location rather than attendance.
For me, "practical days/lessons" doesn't really explain what you're trying to say. I don't associate the word "practical" with language learning. If you told me that someone was learning woodwork, for example, and said they had practical lessons, I would understand that during those lessons they actually worked on pieces of wood, rather than sitting in a chair learning passively from a teacher.