Do you need the Staff of Magnus after killing Arcano?
I sold the Staff of Magnus (despite my misgivings!) because I had found it to be quite useless other than its value in finishing the Eye of Magnus quest. Now I've heard hints that the Eye may cast its gaze upon me again.
Did I render a post-Arcano quest unfinishable by selling off the Staff?
There is an "Aftershock," but you can dispatch the apparitions that appear with regular weapons (and whatnots) and the anomaly will just fade away after they're gone.
Yes, there is a repeatable quest that uses the staff. Its not too important as its a radiant generated quest.
The reward for doing the repeatable is 1000 gold.