The power was granted by the Supreme Court, It is unfettered control of immigration. Hawaii vs Trump. Chrystal clear.
Hardly. Article 1 Section 8 of the Constitution gives that authority to Congress.
Congress has ceded some operational control to the Executive via legislation which they could theoretically recind at any time.
The President does have the ability to exclude certain people on national security grounds but Congress still owns immigration.
For what it's worth, the illegal immigrants are already barred from entry by law - no need for the President to act. The issue is that there is a law explicitly defining a process for those people to request asylum, and it isn't clear that Trump can override that by EO.