The general attitude towards prayers of non-Jews is from Isiah 57.7:
כי ביתי בית תפילה יקרא לכל העמים
-- For My House shall be called a House of Prayer for all the nations.
The Rema paskens that one should respond 'Amen' to the blessings of a non-Jew, providing one heard the entire blessing.
A non-Jew should not say anything false in his prayers, so may not bless שלא עשני גוי, or say אלקי אבותינו (see Shulchan Aruch 199,4 that a convert may say אלקי אבותינו, inferring that a non-Jew cannot). Any prayer which discusses the privilege of being part of the Chosen Nation would have to be edited to be truthful (like בחר אותם rather than בחר בנו, etc.)
- Pesukim paraphrased into the Tefillah
Reciting passages of Tanach found in the nussach hatefillah would not constitute 'Limud Torah' (it is in the context of tefillah, not limud. Similar to the poskim who say one does not need to say Birchas Hatorah before selichos.
The Rambam writes that non-Jews may perform any Mitzvah (with noteable exceptions) and receive reward. Regarding Birchas Hamitzvos, R' Moshe Feinstein (Dibros Moshe Kiddushin §45.2) writes that they should not bless. R' Yisroel Gustman (Kuntresei Shiurim §20) argues and rules that they may bless on any mitzvah they can perform.
Igros Moshe (OCH V2, §25) rules that in an עת צרה, a non-Jew may be obligated to pray for his needs, although in regular tefilla he is not obligated.
According to the understanding of the Shaarei Teshuva and Elya Rabba, the Shulchan Aruch (OCH 55.20) rules that a non-Jew destroys the 'tziruf' of a minyan. (Others translate עכו״ם in that context as the actual Avoida Zara, not just a non-Jew. See Magen Avraham ad loc). The Shevet Halevi (V2, §59.3) paskens accordingly that the minyan should not be in the presence of a non-Jew. Some poskim allow a גר תושב, however.
The Maharil (שו”ת מהרי”ל החדשות סימן כא) writes that one should not allow non-Jews into shul during Birchas Kohanim. The Shu"t Zera Emes (V.3, §12) writes that Kohanim may not duchan if there are non-Jews in the vicinity. Shu"t Yosef Ometz, however, disagrees and permits Kohanim to duchan even in the presence of non-Jews.