What is the ruling on someone who, after rising from the second prostration in the first rak'ah, sat for the tashahhud momentarily out of forgetfulness, then stood up? And is their prayer valid if they did not say the takbir when standing up? And what is the ruling on someone who did that while hesitating about the imam's intention?

The transitional takbir you performed before sitting for the tashahhud by mistake is sufficient for you, and therefore there is no need to repeat the transitional takbir. There is a difference of opinion regarding the prostration of forgetfulness (sujud al-sahw) for an additional, brief sitting in an incorrect place, and a group of scholars have favored not prostrating in this case. The prayer is valid and does not need to be repeated. If forgetfulness occurs while you are a follower of an imam, then sujud al-sahw is not required because the imam bears it for you.