That user is not studying Chinese on Duolingo but is studying other languages. He or she is probably a native Chinese speaker who enjoys learning languages, and there's no harm in that! Furthermore, Chinese characters are learned in South Korea and occasionally used in print alongside the Hangul.
I'm not sure of the grammar on this one. "You're fired" uses the verb "to be fired", so shouldn't there be some form of "해고되다" in there? Or is this just a colloquial way of saying the phrase that doesn't follow a particular grammar? Because 해고이다 couldn't possibly be the verb here, could it?
I'm just guessing based on the grammar and other verbs I know, but if 해고되다 is really 해고 + 되다 where 되다 is "to become", then 해고 would be "fired" in a noun form, right? In which case 해고이다 is "to be fired" (~이다 being the copula/to be). Alternatively, if 해고다 is the verb "to fire" (to terminate employment), then I think that could make 해고이다 a passive voice. But given SeanFogart's response I think it's much more likely the first of my conjectures than a passive verb.