The men knew from studying the stars that the one who would become king had been born. They were trying to learn where he was. Alternate translation: "A baby who will become the king of the Jews has been born. Where is he?"
They were not saying that the baby was the rightful owner of the star. Alternate translation: "the star that tells about him" or "the star that is associated with his birth"
"as it came up in the east" or "while we were in our country"
# worship
Possible meanings are 1) they intended to worship the baby as divine, or 2) they wanted to honor him as a human king. If your language has a word that includes both meanings, you should consider using it here.