1. He's already served two terms as president and isn't eligible to run.
No, he's not eligible to run as president again, but I think he is eligible to run as VP...and then take over as president if something happened to the president. See the discussion in this thread, on the "VP loophole":
https://uselectionatlas.org/FORUM/index.php?topic=75977.0
We might have a conflict between the 12th and 22nd amendment. I know if a president takes over during part of their predecessor's term, then they may serve two full terms after completing their predecessor's term as long as their length of time in office doesn't exceed 10 years.
I don't think it's a conflict. It's something of a loophole in the way the 22nd Amendment is written:
As far as I can tell, there is nothing to stop a person who has been elected to two presidential terms from later being elected Vice President, and then acceding to the higher office upon the incumbent's 'early departure'. By repeating this process, such a person could in effect serve many more than ten years as President...
The 22nd Amendment is all about the rules for being
elected president. It doesn't stop you from becoming and then serving as president via some other means. For example, it doesn't prohibit a Bill Clinton or a George W. Bush, after having served as president for two terms, from being elected VP, and then becoming president via the early departure of the president they're serving with.