Is the rak'ah counted for someone who caught the Imam while bowing, but was unable to bow before the Imam rose, whether or not he said the opening takbir?

In the first scenario, if you pronounce the takbir for ihram and then bow, but the imam rises before you can join him in bowing, then the rak'ah is not considered to have been caught. This is because catching the rak'ah requires reaching the bowing position before the imam rises.

As for the second scenario, if you do not pronounce the takbir for ihram and only pronounce the takbir for bowing, your prayer is invalid. This is because the takbir of ihram is the key to prayer, and one cannot enter into it except with it. Even if you intended to combine the takbir of ihram with bowing, the prayer is not valid according to many scholars.

It is a condition for the takbir of ihram that you be standing. So, if you pronounce it while descending to bow (hypothetically), your prayer becomes a supererogatory prayer, not an obligatory one. This is the preferred opinion of most scholars. There is an opinion that the prayer is valid if you combine the takbir for ihram and bowing, as long as you do not intend only bowing. However, adhering to the opinion of invalidity is more cautious.