(You have an interesting interpretation--I didn't think it went this far.)
The authority of Congress in this matter is only limited by the Constitution (and amendments). The 14th amendment prohibits Congress from removing jus soli. Note the history of the 14th. It was added to expressly make it difficult for Congress to remove jus soli. As such, the 14th would work against #1.
As for scenario #2, yes. Congress can make it so that anyone is eligible to be President. That is, until, via the amendment process, a limitation is placed on Congress’s authority.