The president's executive order "drips with religious intolerance," the full 4th Circuit court in Richmond says in a challenge being watched by some education groups.
Finalists in the National Virtual Supreme Court Competition make their live arguments in an important church-state case that is pending before the real justices.
The justices turned down a case about a student who had disrupted class with "fake burps" that had drawn a dissent in a lower court by now-Justice Neil M. Gorsuch.
The state high court rules unanimously that a warrantless search of an unattended bag at school does not violate the Fourth Amendment.
The president's executive order drew criticism from progressive groups that fear its potential effects in education, including a section ordering the attorney general to issue interpretive guidance down the road.
Judge William H. Orrick refers to briefs filed by California school districts and its state superintendent in support of counties that challenged the president's executive order.
The justices said nominees for a post requiring Senate confirmation may not serve in an acting capacity. That could apply to some Education Department appointees.
In a case being watched by school choice advocates, the justices hear arguments about a state's refusal to provide a playground-improvement grant to a church.
Missouri's governor last week changed the state's position on whether money from a grant program could go to a religious institution, an issue with relevance for school choice.
In a case with implications for school choice, a Missouri church was denied a playground-improvement grant. Now the governor is reversing state policy.