Judaism schismed in the first century.
The Judaism that followed Jesus -- the Judaism of Jesus -- changed its name to Christianity. (Jesus did found the Christian Church -- upon this Rock I will build MY Church.)
The Judaism that rejected Jesus -- Talmudic Judaism -- kept the name Judaism. But it's not the Judaism of Jesus. Jesus is not Jewish in the sense that today's Jews are Jewish.
They can, of course, convert. As can anyone else.
No, and you need to re- read Roman’s 9 to1. Biblical Old Testement Judaism ended at the cross. Messianic Judaism / True Biblical Christianity was then established. You 2nd part is correct. Messianic Judaism is Biblical Judaism. The man you call the first Pope was Jewish all his l8fe.
Modern Talmudic Judaism is descended from the Pharisees.