Do not include in the zimun a Jew who does not fulfil mitzvos that are well known to all Jews (e.g. Krias Shema). Certainly do not include someone who publicly transgresses the Torah. This criterion is more strict than joining such a person to a minyan of 10 to say Kaddish, Borchu etc… (MB2 – BeH “am hoaretz”)
The transgressor himself is obligated in zimun. So if three of them ate together or one is near a group of kosher Jews making a zimun anyhow, they must join in. (BeH “m’zamnin”)
A ger is allowed to be motzi anyone in bentching and say the words “…al shehinchalta laavosaynu…” (ShA4 – MB6)
A woman does not count to be a third person for zimun. However, three (or more) women who ate without men, are allowed to make their own zimun. Women’s zimun should never include the Shem. (ShA6 – MB12,13)
Women who eat together with three or more men are obligated in zimun. If there are three women or more they may break off to form their own zimun group. (ShA7 – MB18)
A boy under bar mitzva cannot be the third person in the zimun. Once he turns 13, not only can he make the zimun, he can even lead the zimun. Since zimun is a mitzva d’rabbonon we can rely on the assumption that the boy now has two pubic hairs. (Rama10 – MB27)
A boy who just turned 13 is not assumed to have two pubic hairs regarding mitzvos d’oraisa. He therefore cannot be motzi other adults in Birkas Hamazon unless that fact is known. (MB27)
A person who is hard of hearing, but can have kavana and understand, can be the third person in the zimun. (Rama10)
A slow-witted person that people consider deranged can be the third person in the zimun if he can have kavana and understand. (Rama10 – MB29)