Is the general public in the USA is supposedly entitled to know whether a given vehicle contains ICE agents? By what legal theory?

This is an inversion of the problem. The general public is entitled to fly drones in many areas and should not be punished just because ICE claims they are operating in an area.

Is there a similar nationwide prohibition on, say, plainclothes police officers?

This is not a valid comparison.

Is there no concern for what would happen in case of mistaken identity?

What does this mean? Why do you think the government should be able to arbitrarily restrict drone operations?

Knowing that a vehicle contains ICE agents, is there a reason that someone should be able to pursue it with a drone? Does this accomplish a legitimate purpose other than tracking the vehicle's position (again, presumably to disseminate the information "this is an ICE vehicle")? Is there a reason why this would not reasonably be seen as harassment from the agents' perspective?

Again, this is an inversion of the problem. If the general public is allowed to operate drones in certain areas, that use should not be subject to widespread, unjustified restrictions.

re ICE agents American citizens, entitled to the same rights as other American citizens?

Most of them probably are citizens.

Do people here believe that the purpose of enacting such no-fly zones is something other than preventing drones from following the vehicles for surveillance and information-sharing purposes? Especially given the idea that the zone moves with the vehicle?

The motivation isn't the problem, the problem is that the implementation infringes on the rights of citizens.

Is there a reason why the government of the USA should not be permitted to enforce its own immigration law? In particular, is there a reason why people who have illegally entered the country per that law, and who have what I'm told is called a "final order of removal", should be permitted to remain within the country?

People opposing the current immigration enforcement regime are not protesting the existence of law, they disagree with the formulation and implementation of the laws. Is it your position that questioning the formulation or implementation of a law should not be allowed?