Is the bowing unit of prayer (rak'ah) counted if the imam bows and rises before the worshipper (ma'mum) completes Al-Fatiha due to seeking refuge (isti'adha), and should seeking refuge be omitted to catch up with the bowing?

Your prayer is valid, and the first rak'ah of your prayer counts for you. You are excused for lagging behind your Imam in order to complete Al-Fatihah. In the future, omit the Isti'adhah (seeking refuge with Allah) and the Istiftah (opening supplication) if you fear that the Imam will bow before you complete Al-Fatihah.