The Ben Ish Chai (Rabbi Yosef Chaim of Baghdad, 1835-1909) records the practice to eat cheese products during Chanukah. This custom serves to commemorate the heroism of a Jewish woman named Yehudit, who killed one of the Greek kings by feeding him cheese so that he would become thirsty, at which point she gave him wine and he fell asleep. Yehudit then killed the king as he slept. Although this incident did not occur during the time of the Chanukah story, it is nevertheless appropriate to bring this event to mind during Chanukah, which celebrates the Jews' triumph over the Greek empire.
The Ben Ish Chai points to a different possible basis for this practice. The Greeks attempted to disrupt Jewish life by issuing a ban against the three institutions of Kiddush Ha'chodesh (maintaining the Jewish calendar, based on the monthly lunar cycle), Shabbat, and Berit Mila. Now the first letter of the word "Chodesh" ("month") is Chet; the second letter of "Shabbat" is Bet; and the third letter of "Mila" is "Lamed." These three letters spell the word "Chalav," milk, and the custom therefore developed to eat dairy products on Chanukah.
On Friday afternoon of Chanukah, the Shabbat candles are lit after the Chanukah candles; women should therefore wait for their husbands to light Chanukah candles before lighting Shabbat candles. This sequence is required both for Halachic reasons, and in accordance with the teachings of Kabalah.
As for the proper sequence on Motza'ei Shabbat, the practice in the synagogue differs from the procedure to be followed in one's home. In the synagogue, Chanukah candles are lit prior to Havdala. Since people generally leave the synagogue immediately following Havdala, very few people would be present for the lighting of the Chanukah candles if it were held after Havdala, thus undermining the element of Pirsuma Nes publicizing the miracle. At home, however, one should first recite Havdala and then light the Chanukah candles.
Summary:
1) It is proper to partake of dairy and cheese products on Chanukah.
2) On Friday afternoon, women should wait for their husbands to light the Chanukah candles before lighting the Shabbat candles.
3) On Motza'ei Shabbat, Havdala is recited in the synagogue only after the congregational lighting of the Chanukah candles, whereas at home, one first recites Havdala and only then lights the Chanukah candles.
