Very interesting questions.
It would seem axiomatic that the U.S. Government can’t deport a natural born citizen of the U.S.
OTOH, the U.S. government is not categorically prevented from doing a two step against a treasonous naturalized U.S. citizen: First stripping them of their U.S. citizenship (e.g., under a theory of fraud) and then unceremoniously shipping them back to the country of which they were originally a citizen.
And the child would be taken with the parents to whatever country they were deported to.
And the child could come back after the age of 35 and run for president, because he was a citizen who did not require any acts to claim his rights of citizenship (unless, I guess, the US denaturalized him, too).