"His own" here means all the Jews, the people of which he was born.
Also, keeping in mind the Jewish mindset and reference points of those to whom Jesus was speaking,
may I suggest that "world" is in contradistinction to Israel, that Jesus came to save people
from all nations of the world.
May I also suggest that those who denied Christ did not have true faith,
as those in Lk 8:13 did not have true faith, and for the same reason,
they had no root, they were not born again, they were just doing "religion." (Mt 7:21-23)
I don't think I understand what you are saying here,
for according to Jn 6:39-40, 10:28-29, Jesus loses none of whom the Father gives him,
and all whom the Father gives him will come to Jesus (Jn 6:37),
and all who come to Jesus and believe in him will have eternal life (Jn 6:40).
Likewise, according to Jn 10:26, the Jews rejected Christ because they were not his sheep;
i.e., not born again,
Peoples from the whole world believe in some kind of god,
but that does not mean rebirth of the Holy Spirit.
Only faith in Jesus Christ is the result of rebirth of the Holy Spirit.