How risk averse are you?
You are on separate tickets so concepts like MCT and connection are not applicable; if you do not show up for your QR flight the whole ticket will be cancelled and QR will not accommodate you on another flight. BA will 100% not check you through to DOH, nor anywhere else on separate tickets. It is almost certain that they won't accept a through check in the return, though there have been mixed reports.
I personally would go out the day before. You obviously bought separate tickets to save money, so you may wish to play it safe by padding in some extra time. Or take a morning flight and explore Berlin a little. The actual arrival/collection/checkin/security doesn't take much time at all at TXL. Your concern is a cancelled or delayed BA flight with limited possibility to accommodate you on the next flight.