a non-Jew can do the prayers of the siddur, make brachot of food or do the service of the Sabbath?
It is nice to see that there are gentiles that want to have a relationship with G-d and pray to him. G-d accepts the prayers of all, in fact when King Solomon inaugurated the First Temple he prayed specifically that when a gentile will come to the Temple to pray, that G-d should specifically answers his prayers. A non-Jew may say a blessing before eating food, but he cannot say the blessings before performing mitzvot, or any blessing that contains the words “ asher kidshanu b’mitzvotav” (such as birkas hatorah, or “al netilas yodayim” because that is like saying a lie in front of G-d, because he was not commanded to do these mitzvot.
Regarding the prayers in the siddur, although there are certain prayers that the non-Jew can say, it is however quite limited, as I will enumerate in part. He shouldn’t say anything that is said because of the special relationship of the Jews to G-d, such as the 3 portions of the shema, or any of the shabbat prayers. He also should not say the bracha of elokai neshoma, as that is referring to a Jewish soul. He will also have an issue with saying shelo osani goy, aved, or isha, because that is a blessing on the amount of mitzvos that one was commanded by G-d to do. He also should not say “ozer Yisroel b’givurah” or “oter Yisroel b’tifara”, as these blessings are referring to teffilin. The bracha of ha’mavir sheno, contain a prayer for success in torah which doesn’t apply to them. Regarding saying the portions of the sacrifices, most of them don’t apply to non-Jews. Regarding saying Boruch She’omar and Yishtabach, I am not sure, since it contains the words “hamehulal b’pe amo” “who is praised by His nation” which refers to the Jews, therefore I would recommend that he not say it because he doesn’t have to say any of the prayers, but he surely should say anything that would not be appropriate for him to say.
He should also not say Ahava Rabba or Ahavas Olam, as these blessings are for getting the torah. He should not say the three portions of Shema as already mentioned. The bracha after shema should also not be said, as it is because of our relationship with G-d. Saying the bracha after shema is also problematic “ezras avosienu ato hu meolam”, you are the savior of our fathers from ever of time”, because of the words “go’al Yisroel” the redeemer of Israel”, which does not apply to a non-Jew. He cannot say any shemona esrei because he cannot say “elokei avoseinu”- The G-d of my fathers.
Therefore, by all means, pray to G-d, but it would be a good idea for you to pen your own prayer, and pray what you would personally like to tell G-d.
Shulchan Aruch O:CH 215-2, Migdal Tzofim 9 Kuntris Ben Noach chapter 9, Poskim.