Q. 1 Can the Gabay of a Shul give an Aliyah to someone who generally keeps Shabbat, but occasionally, not in public and only in need, (not in life danger), turns on the light on Shabbat?
Q. 2 How about when while occasionally traveling and in need, he eats non- kosher food?
Q. 3. Can someone like that be Shaliach Tzibur on a Yortzait?.
Q. 4. If he is a Cohen, can he do Birkat Kohanim?
　
Ans. (1 – 2) An individual that transgresses once or remotely a Shabbos prohibition, and he does so because of the benefit, pleasure or profit attained, (and not out of misguided principles) and not in public, does not become a Mumar, in the sense of being excluded from Klal Isroel. (Shulchan Oruch Y.D. 2- Mishna Berura 55,46 – Pri Megodim 282,7). He can therefore be given an Aliyah at the Torah reading. See Shach (Y.D. 251,1) who asserts that the expression Mumar or Avaryon, suggests a transgression done persistently. The same would clearly apply to a lesser prohibition, such as eating non-kosher food.
Horav Shlomo Miller Shlit”a advised that he should not be called for the reading of any part of the Torah that deals with the particular Mitzvah he violated.

Chasam Soifer (Resp. O.C. 15) maintains that an Aliyah stands today in place of a Korban (see Menochos 110, “Whoever learns the laws of a Chatos is considered as having brought a Chatos”). And Rambam (Maasei Hakorbonos 3,4) states that even when we accept a Korban from a Mumar, to promote Teshuva, we don’t accept a Korban for the offence he actually committed, unless he has already repented.

Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit”a

Ans. (3). Horav Shlomo Miller’s Shlit”a opinion is that although he can be given an Aliyah (see above), and is part of the Minyan, becoming the Shliach Tzibur is more stringent. As the Shliach Tzibur is an emissary and represents the congregants before Hashem, they are entitled to protest if they do not want this individual to represent them. (Even today when everyone prays his own Amidah, the Shliach Tzibur has the potential to say the Tefilah Betzibur together with the late comers) . However we need the majority of the congregants or the elected Gabay to do so in order for the protest to be effective, (In some instances an individual’s protest does count, see Misnah Berurah.55) You may also need, according the case, a Bais -Din or Rabbinical decision, as to if we should believe the testimony of the people who claim he is unfit to be a Shliach Tzibur.

Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit”a

Ans. (4). Rambam’s (Nessias Kapaim 15, 6-7) opinion on a Cohen that is not careful in observing Mitzvos, is that he should nonetheless carry on with the Cohanic blessing, since “we do not say to a person who is a Rosho, increase your evil and avoid performing Mitzvos”. He adds that after all the Brocho emanates from Hashem and not from the Cohen. Mishna Berurah (128,146), mentions that this applies even to severe transgressions, as long as he is not a Mumar on idolatry or the observance of Shabbos.

Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit”a