Are there restrictions pertaining to the recitation of the Oseh Shalom prayer? May it only be said with Kaddish?
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Perhaps this does not fully answer the question, but it was too long as a comment. It addresses an apparent underlying assumption that Oseh Shalom is a component of the amida and kaddish:
The Oseh Shalom prayer, which concludes the amida, kaddish, and birkas hamazon, is appended to the end of those prayers. According to Yoma 53b, a prayer of "shalom" is recited after the amida by way of taking leave.
The specific formulation of this prayer is not found in the Talmud, but it appears in the Siddur of Rav Amram Gaon (9th century), as well as in the writings of several 11th century (and later) rishonim, as following all three prayers. This formulation is founded on the verse in Iyov (25:2): המשל ופחד עמו עושה שלום במרומיו ("He who makes harmony in His lofty heavens").*
The Oseh Shalom prayer is separate prayer recited after the aforementioned prayers. In the case of kaddish, this is indicated by the wording of the Shulchan Aruch (OC 56:5): "After one concludes kaddish, he takes three steps backwards and afterwards says 'Oseh shalom', etc." In this regard, the T'rumas HaDeshen (§13) equates the amida with kaddish. Likewise, interruptions are allowed after the completion of the amida, even before reciting Oseh Shalom (see Mishna B'rura 122:4).
* The Talmud (Rosh HaShana 23b) interprets this verse as describing the carefully separate placement of the celestial bodies. Perhaps this prayer therefore aptly fits the requirement described in Yoma (ibid.) of extending a prayer of peace towards one's left and then towards the right.
As far as I know you may say it whenever you want and there are no restrictions.