The first five minutes of this video make me feel so nostalgic.
I highly doubt that. By definition, nostalgia is the fond recollection of personal events and memories.
Not quite, if you are talking etymologically. It is the pain (αλγος) of returning somewhere.
Yes, somewhere, not anywhere, specifically home since etymologically it is the pain of returning home (νόστος).
Yes, though I think νοστος simply can simply
return; it generally means returning home. (It comes from νεομαι which is cognate to ναιω - or ναιεταω in Epic - meaning
I dwell.)