If a candle extinguished unintentionally, one is not required to relight it, since the lighting is the fulfillment of the mitzvah (Shulchan Aruch 673:2).  However this is only true if the candle had the ability to stay lit for the proper amount of time [i.e. it was set up with enough oil and in a place where it was not likely it would blow out] (Shulchan Aruch 673:2 & 675:2 & Be'ur Halachah 673:2 s.v. im kuvtzu).  If one of the candles went out while one was in the middle of lighting, it is proper to relight it (Be'ur Halacha 673:2 s.v. im kuvtzu).  If one extinguished a candle intentionally, one is required to relight that candle [without a brachah] (Shaar HaTziyon 673:32).  If in the middle of lighting, one of the candles extinguished, one may relight it even though he has not completed lighting all the candles (Koveitz Halachos 5:9).

Beginning 30 minutes before the time to light the menorah, it is prohibited to be eat a seudah [which means eating either hamotzei or a k'beitzah of pas haba bekisnin], learn, or get involved in any melachah which can be very time consuming.  The reason being, Chazal were afraid that one may get distracted and forget to light in the proper time (Mishneh Berurah 672:10).  Additionally, one should not go to sleep (Koveitz Halachos 2:1).

There is a machlokes haposkim if women who are fulfilling their obligation with their husbands' lighting are prohibited from these activities.  Harav Shlomo Zalman Auerbach, zt"l, holds that they are also prohibited (Halichos Shlomo Moadim vol. 1 16:3), however ybl"c Harav Shmuel Wosner, shlit"a (quoted in Shloshim Yom Kodem Hachag pg. 265 ftnt. 42) & Harav Shmuel Kamenetsky, shlit"a (Koveitz Halachos 2:5) disagree.

Preferably one should not light neirosChanukah while it is still day, since it is not as recognizable that he is lighting for the mitzvah.  However, if one is preoccupied [and he will not have any time to light later], then b'dieved one may light from plag haminchah (Shulchan Aruch 672:1).

The Gemara [Shabbos 21b] says that the preferred time to light neiros Chanukah is at "shkiyas hachamah", in order that the candles are lit at the time when people are still outside which creates more parsumei neisa (publicizing the miracle).  In halachah, there are different times that could be known as "shkiyas hachamah"  (see Gemara Shabbos 34b- 35a, Gemara Pesachim 94a and Tosfos Shabbos 35a s.v. trei there are different parts of shkiyah- beginning, end, etc.).  In regards to lighting neiros Chanuka, there is a machlokes haposkim which time the Gemara is referring to.  The different opinions are shkiyah (sunset)(Gr"a quoted in Mishneh Berurah 672:1), 15 minutes before tzaitz hakochavim (nightfall)(Mishneh Berurah 672:1), or tzaitz hakochavim (Shulchan Aruch 672:1).  In Chutz L'aretz there is no universally accepted minhag and there are differing opinions as what to do [i.e. what would be the best time to time to try to be yotzeiall the opinions].  Harav Moshe Feinstein, zt"l, held that one should light 10 minutes after shkiyah (Igros Moshe OC 4:101:6).  Harav Aharon Kotler, zt"l, held 25 minutes after shkiyah (Shu"T Az Nid'baru 7:70).  The opinion of Harav Shmuel Kamenetsky, shlit"a, is to light approximately 20-30 minutes after shkiyah (Koveitz Halachos 3:1 & ftnt. 2).

The Gemara [Shabbos 21b] writes that if someone for whatever reason [i.e. whether on purpose or by accident] did not light the menorah during the proper time, he may [preferably] light until the time of "ad shetichleh regel min hashuk" (that people are coming home- i.e. people are still commonly found outside), because this is still parsumei neisa.  As an aside, this means at the time that most people have finished coming home from work even if they plan on going out again later (Shu"T Teshuvos V'hanhagos 2:342:4).  The Shulchan Aruch [672:2] rules this is approximately 30 minutes after tzaitz hakochavim.  However according to the reasoning that the time of "ad shetichla regel min hashuk" is later, this would affect 2 halachos, both as a leniency and as stringency.  The leniency is that the preferred time to light is extended until people practically are returning home.  The stringency is, one is supposed to have enough oil to last until that time.  Harav Shmuel Kamenetsky, shlit"a, holds there is no obligation to have one's neiros lit until people are returning home, however it is parsumei neisa to have them lit for that long (Koveitz Halachos 3:2).  Therefore, if one is going out to a Chanukah party, as long as it was lit for 30 minutes, one may blow them out even though people are still coming home.