According to this answer, only Congress and the Supreme Court can overturn an Executive Order
reads answer
OK, that's actually a moderately oversimplified and confused answer.
The Supreme Court's authority is derived from Article Three of the Constitution (capitalization in original):
The judicial Power of the United States, shall be vested in one supreme Court, and in such inferior Courts as the Congress may from time to time ordain and establish.
[...]
The judicial Power shall extend to all Cases, in Law and Equity, arising under this Constitution, the Laws of the United States, and Treaties made, or which shall be made, under their Authority;—to all Cases affecting Ambassadors, other public Ministers and Consuls;—to all Cases of admiralty and maritime Jurisdiction;—to Controversies to which the United States shall be a Party;—to Controversies between two or more States;—between a State and Citizens of another State;—between Citizens of different States;—between Citizens of the same State claiming Lands under Grants of different States, and between a State, or the Citizens thereof, and foreign States, Citizens or Subjects.
In all Cases affecting Ambassadors, other public Ministers and Consuls, and those in which a State shall be Party, the supreme Court shall have original Jurisdiction. In all the other Cases before mentioned, the supreme Court shall have appellate Jurisdiction, both as to Law and Fact, with such Exceptions, and under such Regulations as the Congress shall make.
There are a few interesting things to look at here, which I've highlighted in bold.
Firstly, the lower courts have the same "judicial power" as the Supreme Court. The Supreme Court is only special in that its decisions cannot be appealed. So if the Supreme Court can rule something unconstitutional, then the lower courts can, too (but their ruling will probably get appealed).
Secondly, the Supreme Court has "appellate jurisdiction" over the kind of issue we are discussing. Appellate jurisdiction means the Supreme Court can hear appeals from inferior courts, but it cannot handle the case directly and by itself. So it's actually impossible to put President Trump's executive order before the Supreme Court without first going through a district and circuit court (and then persuading the Supreme Court to issue a writ of certiorari, which they are not required to do, and in fact they usually refuse to do so).
Finally, the Supreme Court has ruled that its original jurisdiction cannot be expanded except by Constitutional amendment (see Marbury v. Madison), so there are no statutes or other rules that might allow the Supreme Court to have original jurisdiction over this case.
More broadly, given that the US has 94 district courts, would it be possible to cherry-pick a judge sympathetic to your cause? Would this lead to both sides taking turn going to the District Courts to overrule the other?
Yes, this can happen. If both districts are in the same judicial circuit, one or both rulings would be appealed to the circuit court, which would have to decide who wins. At that point, both the circuit court and all of the district courts under it would be bound by precedent. If the districts are in different circuits, each circuit court would be able to make its own appellate ruling. If the circuit courts disagree with one another, you have a circuit split, which tends to increase the likelihood of the Supreme Court accepting a case. Other positive indicators for Supreme Court review include alleging a constitutional violation and touching on an important area of law. This case does both of those things, so it is fairly likely to come before the Supreme Court even without a circuit split.
