If shares are sold after the ex-date/record date, would eligibility for the rights issue still apply?

Selling shares of the issuer company after the ex-date/record date of the rights issue generally does not affect your eligibility, as the crucial factor is owning the shares on the record date, and the shares would be debited from your account the day after the ex-date. If you sell on or after the ex-date, you still held the shares on the record date, making you eligible for the Rights Entitlement (RE).