Some wine is poured into the kos first so that the wine should no longer be “pagum”. Any time one drinks from a kos, the remaining wine is pagum and should not be used for a subsequent kiddush. Pouring some more wine in “fixes” the pagum wine.
For kiddush, the person who said kiddush should drink a “m’lo lugmov”, but if he can’t, someone else can.
For havdalah, one must drink a r’viis, to avoid a shailah about a bracha acharona. Actually, one should never drink more than a “m’lo lugmov” but less than a r’viis for the same reason.