Well this is my theory: The demons had already taken over the human world(unlike the previous games) and Mundus being the king also wanted to take over the Angel realm. However, he knew that ordinary demons weren't strong enough , so he sent his general Sparda to capture a relatively powerful angel and impregnte her to create a strong hybrid. Reluctantly he captured EVA but being a demon of high morals and pride could not bring himself to carry out rape. Instead he gave her shelter (the angel community might have regarded her as tainted and would not accept her into their society) and eventually they fell in love. However mundus was enraged so Sparda fled with his family. But they knew that Mundus will eventually find them so he and Eva abandoned their children at an orphanage to keep them safe, While they ventured into Mundus's fortress to bring about an end to his tyranny so that Dante and vergil could live a normal life . However they failed. Years later mundus plans to create another hybrid but this time he does the impregnation himself. Lilith might be a fallen angel and their son might turn out to be the reboot version of nero.
Vergil might be looking to complete his mssion of taking over the angels but he wants to do so as the king so he affiliated himself with the order to kill Mundus and become the leader. Eventually going to war with the angels.
Trying to answer ur questions:
1)Maybe he didnt find him, or he was hoping he could use him
2) Same question here, I think Sparda is still alive but dormant
3) He must have had some mentor who presumably was killed by demons
4)Ive mentioned it in my theory
5)To disguise themselves (humans dont really know about demons)
Hope u like my theory