Would putting også in front of er be grammatically correct? I feel like doing so would indicate whether you are saying "we also are students" (as in this group is students in addition to another group being students) or "we are also students" (as in we are, among other things, students).
Depends on your interpetation of "gramatically". I would say no, assuming grammar includes typically correct word order. Typically adverbs follow the verbs when Danes speak. If you place an adverb in front of a verb, chances are they will still easily understand you, but it sounds better for them if the adverb comes after the verb. Its like some saying something like "i drink always water" in english. It makes sense, sounds a little weird, the meaning is intact, but the grammar is debatable.