If someone has never kept shabbos, works on shabbos or owns a business that is open on shabbos, can he be counted towards a minyan?
Is there any leniency if the minyan is short by one, and they get anyone from outside to answer Amen for Kaddish?
Historically a non-shomer Shabbat Jew did not count as part of a minyan, however in recent generations, a number of halachic decisors taught that the status of Shabbat-desecrators is different today. They do not desecrate Shabbat to defy God but as part of a general movement of non-observance. As such many prominent poskim agree they can be counted. See R Melamed below or OU (here). R Moshe Feinstein has a number of tshuvot where he allows it in case of need (see here).
R Eliezer Melamed (here) describes it well
A Jew who has sinned, for example, by eating forbidden foods, committing adultery, or transgressing other biblical commandments, is still counted as part of the minyan. Despite the fact that he sinned, in his inner core he surely desires to be a part of the holy objectives of Am Yisrael. As the Chachamim say (Sanhedrin 44a), “A Jew, even one who has sinned, is still a Jew.” However, a person who sins in order to purposely incite God’s wrath does not count as part of a minyan, since he himself demonstrates that he does not belong to the Torah or to Israel (Mishnah Berurah 55:46-47).
According to some poskim, one who desecrates the Shabbat in public, even if he does it for his own pleasure, is considered an idol worshiper and does not count as part of the minyan (Mishnah Berurah 55:46). However, in recent generations, a few prominent poskim have taught that if a person who desecrates the Shabbat wants to be part of a minyan, we can include him. This is because today, the status of one who desecrates the Shabbat is different. In the past, when all of Israel observed the Shabbat, anyone who dared to publicly desecrate the Shabbat, even if he did it for his own pleasure, outwardly defied all of Israel and was therefore deemed as one who sins intentionally to arouse anger. However, in recent times, when unfortunately many Jews do not keep the Shabbat, its observance is not a measure of a Jew’s identification with his heritage, and therefore, if he himself wants to be part of the minyan, he may be counted. Still, it is not proper to appoint him as chazan.
Rav Aharon Feldman, Rosh Yeshiva of Ner Israel in Baltimore, was once returning to Baltimore on an early morning flight. He was leaving too early to daven with a minyan before he left, and he was not going to make it to Yeshiva in time for shacharis at Yeshiva. Therefore, it was arranged for nine bochurim to not daven at the Yeshiva minyan in order to make a minyan with the Rosh Yeshiva. However, one of the bochurim forgot and davened with the Yeshiva minyan.
They pulled aside one of the janitors of the Yeshiva who was a non-practicing Jew, and wanted to count him for the tenth. The Rosh Yeshiva asked him 3 questions - does he believe in G-d, did G-d give the Torah, and is there reward and punishment. The janitor answered in the affirmative, and so they counted him for the minyan.
So, the Rosh Yeshiva held that a non-observant Jew could complete his minyan.
Source: one of the bochurim who was in that minyan told me about it later that day.
Cross-referenced from here