Actually, No he doesn't. The Constitution makes HIM a co-equal branch of the government which means the judiciary cannot dictate to him about anything. Ties between the executive and judicial branches are determined by the Congress through proper legislation. I wish Trump would act as a President rather than an ordinary citizen when it comes to the orders of the courts.
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Not sure what that means. The executive can act on explicit or implicit powers in the Constitution, or by legislation. The courts have to decide what is Constitutional, or implied, or legislated.
OTOH, there's nothing in the National Emergencies Act about allowing the courts to decide what is an emergency and what is not. That is entirely up to the President and Congress. Congress would need enough votes to override a veto if they want to terminate the emergency. But the emergency has to be declared on some legal authority. In the past the authority was somewhat circular: e.g. https://www.law.cornell.edu/uscode/text/10/2808 used by GWBush after Sep 11. Bush had already declared the emergency by EO: https://georgewbush-whitehouse.archives.gov/news/releases/2001/09/20010914-4.html and several more orders after that to help sieze assets and stop terrorist financing.