By the way, I wonder how and when Michael found out that Roth was still alive since he didn't inquire when he arrived back in the US.
Tom told him--first thing after he asked Rocco and Neri to step outside.
I know. However, Tom's info about Roth was presented as incidental and as a response to a question about Fredo. I would think that Michael would surely want to know if his plot had succeeded. Remember, he yelled to Fredo that Roth was dead. To find out that he was still alive would seem to be a major concern.