On Erev Shabbos, if a woman is running late to light Shabbos candles on time, she is permitted to light the Shabbos candles immediately, once her husband has lit even one of the lights on the Chanukah menorah, and she will be yotzeh with her husband’s lighting.
(Rama end of 671:7, Halichos Yosef 679:1, Ben Ish Chai 1:Vayeishev:Chanukah:20)
The Mishnah Brurah brings in the name of the Pri Migadim and Shaarei Teshuva that on Erev Shabbos, lechatchila, one should daven Minchah before lighting Chanukah licht (if possible). Nevertheless, most communities are not makpid on this and daven after lighting.
{Shulchan Aruch w/Mishnah Brurah 679:1 see Shaar Hatziyun 7, see Shaarei Teshuva, Piskei Teshuvos 679:2, Aruch Hashulchan 671:26)
If one has a choice of davening Minchah before lighting without a minyan or davening after lighting with a minyan, one should daven after lighting with a minyan (Piskei Tshuvos 679:2 citing the Shela Hakodosh, Elya Rabba, S’U Divrei Moshe 1:15).
The generally accepted minhag (which follows the Taz) is that on Motzoei Shabbos, at home, one should first make Havdalah and then (promptl) light the menorah (Kitzur Shulchan Aruch 139:17, Sdei Chemed Chanukah:19, Aruch Hashulchan 681:2, see however SA w/Mishnah Brurah 681:1,2 see Biur Halacha there, Piskei Tshuvos 681:2).
On Motzoei Shabbos, it is best for one (if possible) to re-use the candle that was used for Havdalah for the Chanukah Menorah; since a mitzvah was done with it, one should use it for another mitzvah (Shulchan Aruch w/Mishnah Brurah 681:1).
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