Using the flame of the Shabbos candles to light another candle?
It is forbidden to light a mundane candle/match from a Shabbos candle [even if one did not yet accept Shabbos with this lighting]. [Thus, one may not use it to light the gas, or a cigarette and the like.] Furthermore, the custom is to forbid doing so even if this candle will be used to light the other Shabbos candles. [Thus, if the match, or candle used to light the Shabbos candles, extinguished, it may not be relit from an already lit Shabbos candle even if one did not yet recite the blessing.] However, it is permitted to light one Shabbos candle directly from another Shabbos candle, and so is the custom. Nevertheless, it is initially proper to designate a candle to be used to light all the other Shabbos candles, rather than to light them from each other.
May one melt the bottom of a Shabbos candle using an already lit Shabbos candle?
One is not to do so due to it being belittling of the Mitzvah of the already lit candle.
 Hisorerus Teshuvah 1:160
 Michaber 154:14 regarding Shul candles; 674:1 regarding Chanukah candles and 674:2 extends the ruling also to Shabbos candles; Mishneh Berurah 263:4; All Poskim in coming footnotes! See Shabbos 22a
 1st opinion in Michaber 674:1 and final ruling of Rama ibid regarding Chanukah candles; P”M 674 A”A 1; M”B 263:4 “The custom is to be stringent”; 674:5; Biur Halacha 263 “Shtei Pesilos” explains that one may not light the Shamash to light the other candles.
Other opinions: Some Poskim rule one may light a mundane candle for the sake of lighting other Chanukah candles, so long as there is no concern that it will extinguish prior to having a chance to light the Chanukah candles. [2nd opinion in Michaber ibid; Sefer Haterumos; Ran; Shmuel in Shabbos ibid] The same leniency would apply regarding the Shabbos candles.
 Piskeiy Teshuvos 263 footnote 85; However, see Nishmas Shabbos 1:329 that based on the Levush and Taz 674:1, if the candle is designated only for lighting the Shabbos candles, as is common today to purchase a silver candle extension which is used only for lighting the Shabbos candles, then it may be relit from an already lit Shabbos candle.
 Darkei Moshe 263:2 in his understanding of Hagahos Maimanis and Mordechai, brought in M”A 674:1; Implication of Rama 674:1 “It is customary to be stringent by Chanukah candles”; Chemed Moshe 263:2 “One lights Shabbos candles from each other and so is the custom”; Chayeh Adam 154:26 “On Shabbos we light from one candle to another”; P”M 674 A”A 1 that so applies even according to M”A ibid; Mishneh Berurah 263:4 “It is permitted to light from one candle to the next”; 674:5; Biur Halacha ibid “And so is the witnessed custom”; Kaf Hachaim 674:9; Piskeiy Teshuvos 263 footnote 73
The reason: As all the Shabbos candles serve the Mitzvah, and hence lighting them from each other is not considered a belittlement of the Mitzvah. [M”B 263:4; See Biur Halacha ibid for explanation]
Other opinion: Some Poskim rule it is customary to be stringent not to light one Shabbos candle from another Shabbos candle, even directly. [M”A 674:1 in his understanding of Mordechai, Hagaos Haram Tiktin and Tosafus that “The custom applies by all Mitzvah candles”] The Poskim ibid negate this understanding of the M”A. [Chemdas Moshe ibid; ] See, however, P”M ibid, brought in Biur Halacha ibid, who explains that perhaps even the M”A ibid only intended to be stringent regarding lighting through a Shamash, however, to light directly even he would agree that it is permitted, and hence there is no dispute. Nonetheless, he concludes that it is better to light from a Shamash then to light directly.
 Elya Raba 263 in name of Paneiach Raza; Biur Halacha 263 “Shtei Pesilos”; P”M 674 A”A 1
 SSH”K 44:38 footnote 187; Piskeiy Teshuvos 263:7