For the record:
1. Legal residents (permanent and otherwise) have constitutional rights. Numerous citations. (Beldin: "wishful thinking by leftists who want to ascribe U.S. Constitutional rights to non-U.S. citizens.")
2. Illegal aliens have constitutional rights. Plyler v Doe (Beldin: "foreigners who do not possess U.S. residency authorization ... do not enjoy ANY of our rights under The United States Constitution.")
3. Refugees/asylum seekers have rights under US and international law. INA 1158 and 1231(b)(3), Foreign Affairs Reform and Restructuring Act 1231, Geneva Refugee Convention 1, 3, 4, 16, 33 etc.
4. Other visa applicants may or may not have rights under US law, depending upon their status and relationships.
The president has wide discretion under INA but he has to abide by INA as a whole, other relevant laws, the US constitution (powers and application) and the US' international obligations.
Complainants are presenting their cases in order to seek to expose unlawful elements of the EO.
The courts decide if elements of a law or an executive order are unlawful or unconstitutional. They tend to defer on immigration where national security is in issue. But this doesn't mean they will permit the president to claim powers which INA does not confer. If he defines a final solution for Moslems under an immigration EO, I imagine the Supreme Court will say no, INA doesn't give you those sorts of powers.
This isn't hard. But it is over.