Not possible in the US. Musk isn't a born American but a naturalized citizen. He can never be president without a constitutional amendment and that isn't going to happen ever.
I assume that also means a naturalized citizen has to be VP and Speaker of the House, as that’s the order of who becomes president if the higher up one dies? Or would we have a new election if it ends up being an ineligible individual?
You're remembering incorrectly. Schwarzenegger, while elected governor of California, never made a run at the White House. As the other poster noted, Schwarzenegger couldn't do that even if he wanted to.