1 – When a goy converts (ger) to Judaism, which tribe does he belong to? How is the tribe of a ger designated?
2 – a son of a Jewish mother and a non-Jewish father, which tribe will he belong to?
Belonging to a tribe is something that can only go from father to son, therefore a ger doesn’t have a specific tribe. Even a Jew that has a non-Jewish father does not have a specific tribe. This is what happened in the desert, with the person who cursed. The reason he cursed was because he was told that he has to leave the place that he wanted to stay with the tribe of Dan, but he was told that he doesn’t have the right to stay there.
On the other hand a ger should not be too concerned with this, as the Prophet Isiah (56-3,6-7) “And let the son of the gentile not say, alas Hashem has separated my from His nation… And to the sons of gentiles (geirim) who accompany Hashem, to serve Him and be His servants… And I will bring them to my holy mountain, and I will gladden them in My house of prayer”. Therefore, the geirim should know that when Moshiach comes, although they would be part of a specific tribe, but Hashem will take care of them.
Rashi Leviticus 24-10