Let’s look into some of these claims. If you think my assessment is wrong, please let me know. I’ll be happy to learn more about these things.
1) The president has a right to make that order. TRUE.
2) Border agents have a right to deny entry to anyone for any or no reason. TRUE (unfortunately).
3) Law enforcement agents can deport any legal immigrant from US soil at any time for any or no reason. FALSE. I’m not sure whether this is what you mean by “people not having the right to stay,” but it is false as far as I know. Legal temporary residents (US visa holders) have the right to stay, once in US soil, until their documents expire or are revoked. Otherwise they may only be removed with a cause (criminal offence for example).
4) The courts are wrong to even question the president on this matter. FALSE. Although some of the president’s actions are not justiciable (such as some foreign policy), cases involving and affecting legal [temporary (foreign student, worker, etc) or permanent] residents of the US are justiciable. So they can be judged by the courts. And even if the president has the right to submit his order, the courts may still find the actions unconstitutional because of the damage it exerts on thousands of lawful immigrants (visa holders in this case, since green card holders are exempt).
As I said please let me know if you think any of these are false. Thanks!