Did I err in hitting my twin brother who impersonated me and engaged in an illicit relationship, which led to his traveling, and am I the cause of his travel?

A Muslim is not permitted to assault his Muslim brother by hitting him. The proper course of action was to advise him and remind him of Allah. If he did not respond, then his matter should have been referred to whoever has the right to discipline him, such as his father or the judiciary. You must seek forgiveness for what you have done and ask your brother for his pardon. Since he has forgiven you, we hope that you bear no sin. You should continue to treat him kindly and advise him gently if he persists in actions that violate Islamic law.