Are Bnei Noach normally allowed to take part in any of the liturgy in Orthodox synagogues? For example: responding to Torah blessings, singing Adon Olam, reciting Kaddish (except Mourner's Kaddish), etc. Obviously there would be no Aliyah. What are normally the rules regarding Noachide participation?
After seeing the specific question asked at the end of the comments section, (see there) i see no reason why a Ben Noach would not be allowed to join in Adon Olam or answering Amen to any blessing, which is like saying "I agree."
Answering Kaddish seems to be the same. In Aleinu, we look forward to the messianic times when every person will accept the Oneness of haShem. Why not say it a few years (or days!) early?
As for saying those parts of davening (prayer) that come from Tehillim (Psalms), I think it would be alright.
As for making some blessings, it would seem not. Certainly (1)those that say "kiddishanu"(He has commanded us) or (2)in the Amidah, which starts with "...G-d of our fathers," which would not be true. This has been discussed in responsa and Halacha regarding (1) a woman taking a lulav&etrog; &(2) a Ger Tzedek & has been decided that (1) the custom has been established that women do this mitzvah so she can say "He commanded us" and (2) a Ger is like a newborn (Jewish) child and should say "Our father"(and not skip the phrase). Both these don't apply to a non-Jew. It would seem that a non-Jew could say blessings that are praises (for food, Asher Yatzar, most morning blessings, etc.)
Rav Moshe Feinstein, Ztz"l, Igros Moshe, 2/25, states that a non-Jew is not commanded to pray but if he chooses to pray, he would receive reward like one who is "eino Metzuveh v'oseh,"(not required but volunteers). With that in mind, saying the Amidah seems appropriate as long as one would skip the phrase "G-d of our fathers."