(This is not totally clear cut. Controlling entry across the border is an issue uniquely within the President's powers. I think the due process issue overrides that for immigrants who have been granted visas and green cards, but I could see Thomas being "unitary executive" enough to see differently.)
If the supreme court starts upholding executive orders over standing law, I'd be terrified.
The law as written references a lot of other sections, but it seems pretty straightforward. My interpretation is that Trump's order is illegal.
I'm going to defer with the ACLU's assertion that it's unconstitutional/illegal because I trust them and they know a lot more about the law than I do. Here's their explanation for why they think it's illegal: https://www.aclu.org/blog/speak-freely/well-see-you-court-wh...
As an equivalent example of this, if the court rules a particular law unconstitutional and congress passes the same law under a different name, that law would be utterly powerless as well. In general, Changing the name of something doesn't change its legality.