Could a Ben-Noach actively practice a religion other than Judaism as long as it isn't considered avoda zara? Suppose, for example, he was born Muslim. Could he continue to actively follow Islam and still be considered to be following the Noachide laws?
While I do not mean to suggest that they cannot have a portion in the world to come, if they do keep the 7 mitzvos, they are nevertheless not permitted to observe another religion:
The general principle governing these matters is: They are not to be allowed to originate a new religion or create mitzvot for themselves based on their own decisions. They may either become righteous converts and accept all the mitzvot or retain their statutes without adding or detracting from them. Rambam, Mishneh Torah, Hilchos Melachim 10:9
Yes, a Ben Noach is allowed to practice another religion as long as it doesn't break any of the 7 Noahide laws.
As for your example, according to the Rambam Muslims have no problem with their belief system (obviously there are other issues that the Rambam does have - but that is not within the context of this discussion).
Rabbi Tzvi Freeman of Chabad.org has suggested that Noahides should draw on aspects of the traditional faith in which they were raised. For example, see here:
Under this line of thinking, a Noahide with Muslim roots should not practice Islam exactly, but a Noahidism with Islamic influences.
For example, perhaps post-Muslim Noahides will one day publish a volume with selections from the Koran, hadiths, etc., while leaving out anything that contradicts Judaism (such as the Muslim belief that the Jews edited the Torah to take out prophecies of the prophet Mohamed) or speaks negatively about Jews.