The Mitzvah of Kiddush from the Torah
1. There is a mitzvas asei deoraisa to verbally sanctify the day of Shabbos when it arrives, as the pasuk says, “זכור את יום השבת לקדשו” – meaning, mention words of praise and sanctification about Shabbos. Miderabanan this should be done over a cup of wine (רמב''ם פכ''ט שבת).
2. Shabbos day. According to most poskim, Kiddush on Shabbos day is only derabanan out of respect for the seudah (מ''ב סי' רפ''ט סק''ג).
Women
3. Since whoever is obligated to keep Shabbos is obligated to mention Shabbos, women are also obligated in Kiddush (שו''ע סי' רע''א ס''ב).
4. Arvus and women. The leading poskim argued whether women are included in arvus [a concept that allows being motzi others] (רא''ש ורבינו יונה פ''ג ברכות כ' ע''ב, ריטב''א הל' ברכות פ''ה ה''ב מרדכי מגילה סי' תשצ"ז ח'). The Mishnah Berurah and other poskim imply that they are (סי' רע"ג סק"כ ותרע"ה סק"ט), and thus one can be motzi a woman even if he already fulfilled the mitzvah. Also, women can be motzi other women.
5. However, others hold arvus doesn’t apply to women (דגול מרבבה סי' רע''א, שו''ת חת''ס סי' כ"א וקמ"ג). Thus, lechatchilah it is best to take this opinion into account when possible and one should not be motzi women in Kiddush if he already made Kiddush unless he is also being motzi men at the same time (שו''ת מנח''י ח''ג סי' נ''ג נ''ד, שו''ת אג''מ ח''א סי' ק''צ).
6. A woman being motzi a man. Although strictly speaking a woman can be motzi a man in Kiddush according to the primary opinion, that arvus applies to women, a woman should refrain from being motzi men who are not family members in Kiddush (מ''ב סק''ד). It seems that it is disrespectful to the mitzvah for a woman to be motzi even her husband or males in her family who are gedolim (שו''ע הרב ס "ו). However, if a man cannot drink wine or grape juice, it is better for his wife to make Kiddush on wine and be motzi him than for him to make Kiddush on bread (כ''ק מרן גאב''ד בעל שערי טוביה).
