Is the رکعه counted for one who catches up with the Imam while he is bowing, then says “Allahu Akbar” and bows before the Imam rises, even if he does not complete the obligatory glorifications? And is the prayer of one who does not stand completely upright after bowing, but remains in an arched posture, invalidated?

If the worshipper joins the Imam while he is bowing, he says the Takbirat al-Ihram (opening takbir) while standing, then bows immediately. The Takbirat al-Ihram is sufficient for the takbir of bowing, provided that the Takbirat al-Ihram is said while standing. If he says the takbir while descending into bowing, his prayer is not valid as a fard (obligatory) prayer. If the worshipper does not straighten completely after rising from bowing, and remains slightly bent, his prayer is valid, and his straightening is sufficient as long as his bending does not reach the extent of bowing.