When may a husband forsake his wife, then strike her with a non-painful beating? And is he sinful if he does not forsake her when she deserves to be forsaken?

When a wife shows 'nushuz' (disobedience/rebelliousness), the remedy for it is, first, admonition. 'Nushuz' is when the wife elevates herself above her husband and disobeys him in matters where she is obligated to obey him concerning marital life. The husband is not required to abandon his wife when she shows 'nushuz' if admonition is ineffective; rather, it is permissible for him to do so. The general principle is that abandonment is forbidden without a legitimate شرعي (Shar'i) reason; otherwise, the wife would be permitted to seek a divorce because of it.