Regarding 1, I would be shocked if you could not. Universities only will check about your nationality if you are trying to claim a discounted fee normally available to EU citizens, otherwise they do not care at all about you (I did a post graduate diploma and was never asked anything about my nationality or migratory status).
Regarding 2 is trickier, doesn't your visa say anything that may shed some light? You should simply ask the authorities, no need to beat around the bushes.
Regarding 2 is trickier, doesn't your visa say anything that may shed some light? You should simply ask the authorities, no need to beat around the bushes.