Is it permissible to bow with the Imam without reciting Al-Fatiha, especially since whoever does not recite Al-Fatiha has no prayer, and what should the latecomer do if he is unable to complete the recitation of Al-Fatiha before the Imam rises from bowing?

If you enter the mosque and the Imam is bowing (in ruku'), then say the takbirat al-ihram while standing, then immediately bow. In this case, the raka'ah will be counted for you, and the recitation of Al-Fatiha is waived. However, if you catch the Imam standing (in qiyam), then reciting Al-Fatiha is obligatory for you. If the Imam bows before you complete it, then finish its recitation and then follow him.