doesn't work. White throws in at
and Black ends up with two liberties against White's three. (
at
)
If White cuts at b, does work, because White b takes away a liberty.
As for the third question ...
If -
and a secures White's big group in sente or b captures Black's large group. White is ahead in the game, so securing the group is enough.
If at b - which is good style - White c both threatens to capture the cutting stones or to get ahead in the aforementioned capturing race.