May one use the oil that remains after the candles have extinguished?
The leftover oil of the 8th day’s Chanukah lighting is to be burnt on its own if the candle extinguished prior to burning for a half hour [past nightfall]. [This law likewise applies for the leftover oils of all the other nights, although regarding the oil of the other nights one may simply reuse the leftover oil for the lighting of the next night. However regarding the oil of the 8th night, the leftover oil must be burned on its own due to it being the last night of Chanukah.]
Oil that remains after it was lit for a half hour: If the candle contains more than 30 minutes worth of oil, then the first thirty minutes of oil [lit past nightfall] is considered designated for the Mitzvah, while the oil that remains after these thirty minutes does not contain any Mitzvah status, and may be even initially used for any purpose. [However some opinions rule that all the oil is designated for the Mitzvah and thus may not be used for any other purpose. According to all, if one stipulated on his oil that only the oil which burns for the first half hour is designated for the Mitzvah, then the remaining oil may be used for other matters. Practically, it is initially best to suspect for the latter opinion and stipulate upon placing in the oil that all oil that remains past a half hour is not sanctified for the Mitzvah. If one explicitly placed more than a half hour worth of oil for the sake of the Mitzvah, then all the leftover oil must be burnt on its own after Chanukah.]
What should one do with the leftover wicks of the Menorah: The leftover wicks of the Menorah are likewise to be burnt.
If this oil mixed with other oil: If the leftover Mitzvah oil became mixed with other oil, it is nullified in 60x. If the mixture does not contain 60x then the entire mixture is forbidden in benefit and is required to be burnt on its own. Some Poskim rule one may not add more oil to the mixture in order to nullify it in 60x. [Other Poskim however rule that if one has more oil, he may add it to the mixture until it is nullified in 60x.]
After Chanukah, the leftover oils of the first half hour, and the wicks which were used to light the Chanukah candles, must be burnt on their own. It is forbidden to use them for any other purposes. It is proper to stipulate before Chanukah that the oil which remains after the 30 minutes of lighting is not considered designated for the Mitzvah, and hence does not require burning after Chanukah according to all.
Must one burn the leftover oil that remains in the bottle? May this oil be used for other purposes?
May the leftover oil of the first half hour be used to light Shabbos candles?
 Michaber ibid as explained in M”B 677/18 that “Hatzarich Leshiur Hadlaka” means that there was enough oil to last 30 minutes and it extinguished before that time; So also explains Levush; Kaf Hachaim 677/26
The reason: As the oil of the half hour of burning as designated for its Mitzvah. [Michaber ibid; See Ramban Shabbos 21b]
 Tur; Levush; M”B 677/17; Kaf Hachaim 677/26
 Michaber ibid as explained in M”A 677/10 and M”B 677/18; Kaf Hachaim 677/26-27; See also Michaber 672/2
 See Biur Halacha 671/1 “Ubilvad”
 Mahariy brought in Beis Yosef 677; Bach and Tzeida Laderech, brought in M”A 672/4; 677/10; Peri Chadash 677; brought in M”B ibid; Kaf Hachaim 677/26
 M”B 677/18; Poskim in next footnote
 M”A 677/10; Elya Raba 672/2; Chayeh Adam; Poskim brought in M”B 677/18; 672/7; Kaf Hachaim 672/21
 Levush; Kaf Hachaim 677/26
 Kitzur SHU”A 139/20; Tur; Beis Yosef; See Kaf Hachaim 677/26 in name of Levush
 Michaber 677/4
 This refers to the oil that remains when a candle extinguished prior to a half hour after nightfall, whether of the 8th night, or of any other night of Chanukah, as explained above. However, the oil that remains after a half hour after nightfall, is not considered Mitzvah oil according to the many opinions, and hence Bedieved if the oil mixed, seemingly nullification is not required, even if a stipulation was not made.
 Turn in name of Mahram of Rothenberg
 It is not similar to the law regarding adding more wood to forbidden wood on Yom Tov, being that the benefit of the wood is only received after it turns to ash. [M”A 677/12; Kaf Hachaim 677/30; See Admur 507/3
 Yoreh Deah 99/6
 M”A 677/11; Tur 677; Levush; M”B 677/19; Kaf Hachaim 677/29
 The reason: As we suspect that one may come to use it in the interim, either for eating or for lighting. [ibid]
 Implication of Michaber ibid; Piskeiy Teshuvos 677/9; See Aruch Hashulchan 677/6; Chochmas Shomo 677; Shraga Hameir 5/118