What is the status of encouraging non-Jews to observe the Sheva Mitzvos and the relevant beliefs which they are obligated in? Is there any type of prohibition involved (many argue as if missionaries are wrong simply for trying to get others to change their beliefs rather than because they promote false beliefs)? Is there a mitzvah to do so provided one does so with proper derech eretz (i.e. not badgering/harassing people or befriending them under false pretexts to do so)?
Similarly, is there any obligation or mitzvah for an individual to discourage a non-Jew from converting or is this obligation only relevant to the Beis Din/Rabbi?
There is certainly no prohibition in encouraging a non Jew about the Sheva Mitzvos. This is certainly a meritorious act as it is furthering the fulfillment of Hashem’s will in this world, which is certainly for all gentiles to be keeping the Sheva Mitzvos.
The obligation to discourage a convert would seem to be only on the Bais Din whom he approaches to ask to convert him, see Shulchan Aruch Y:D 268:2.