The Gemara [Megillah 32a] says "If 10 people have read from the Torah, the greatest one of them should receive the honor to be go'lel (roll) the Sefer Torah". The Gemara continues "the person who rolled the Sefer Torah is mi'kabel sechar keneged kulum (receives reward equal to [the reward of] all them"). Tosfos [ibid s.v. gadol] explains that since the greatest reward is received for performing gelilah [i.e. it is the most honored part of krias hatorah], it is therefore befitting that the most honored person receive this honor.
There is a machlokes as to what exactly the Gemara is referring to when it writes "ha'golel". Some opinions hold that it is referring to the galilah before the Torah reading [i.e. the one who rolls the Torah to the proper place], and others understand it is referring to what we know as [hagbah and/or] galilah (see Sefer Tehillah L'Yonah Megillah pg. 399).
The Mishneh Berurah [147:5] understands the Gemara is referring only to the person who does the hagbah [but not galilah]. However, the Chazon Ish zt"l and ybl"c Reb Chaim Kaneivsky, shlit"a, understand that it is referring to both the person who performs hagbahand the person who performs galilah (Orchos Rabbeinu 3:pg. 216).
The Aruch Hashulchan [147:8-9] asks why should one who performs gelilah receive so much reward; what was "so great" about what he did? He suggests that perhaps the reason is since he closed the Sefer Torah, thereby avoiding the Torah being left open.
The Shulchan Aruch [147:1] paskins the greatest person in the minyan should receive hagbah. However, the Mishneh Berurah [147:7] writes that nowadays the minhag is not this way; rather we give hagbah to different people etc. because of darkei shalom (keeping peace). The Mishneh Berurah continues, someone who physically finds it difficult to do hagbah [and will not be able to lift the Torah for a long enough time], the gabbai should not offer the honor to him, and if he is offered he should not accept it. As an aside, the implication of the Mishneh Berurahis that unless one has physical difficulty etc. lifting the Torah, one should not refuse this special opportunity.
Similarly, the Shulchan Aruch [147:1] paskins the greatest person present in the congregation should get the honor of galilah. The Mishneh Berurah [147:7] explains that this is only m'ikar ha'din, however the minhag is to give galilah even to ketanim who are old enough to understand the importance of devarim she'bikedusha, in order to educate them how to perform mitzvos. However, one should not give the honor of galilah exclusively to ketanim, for it diminishes the importance of the mitzvah (Halachically Speaking 2: pg. 185 quoting Siach Tefillah 13:2:2).
The Gemara [Shabbos 14a & Megillah 32a] teaches, whoever touches a Sefer Torah with his bare hands loses the reward of the mitzvah he performed. This is even if one washed netilas yadayim beforehand (Rama 141:1). The Levush [OC 147] explains, the reason why is it shows a level of disgrace and light headiness in front of a Sefer Torah. Practically, if one touched the parchment while performing galilah, he would not receive reward (Mishneh Berurah 147:1).
It is permitted to touch the parchment of a Sefer Torah for the purpose of the SeferTorah if it is impossible to accomplish what needs to be done without touching [e.g. fixing a ripped Sefer Torah] (Mishneh Berurah 147:1). However, since gelilah one can use a tallis [to lift up the drooping parchment], one needs to be careful not to touch it directly (Mishneh Berurah 147:3). One may touch the handles of a Sefer Torah directly (Mishneh Berurah 147:2).