Actually one of the areas they have the most immunity. Bush passed the "American Service-Members' Protection Act," authorizing unlimited use of force/resources to free any US personnel being detained by the ICC.
https://en.wikipedia.org/wiki/American_Service-Members%27_Pr...
They don't need immunity from ignorance, they have a blanket immunity.
As for the case you cited, it was a civil suit, therefore ruling only covers civil law, not criminal. That said, does not protect "the plainly incompetent or those who knowingly violate the law". [1]
All Presidents are fully aware by law their powers are limited, no reasonable official would claim otherwise. No reasonable person or official would claim their actions were legal, hence why they knowingly hid them from FISA & Congress.
[1] Malley v. Briggs, 475 U.S. 335, 341 (1986).
The President does not have the power to fire federal prosecutors serving in subsequent administrations. It's true that you've provided the explanation for why (even without the DoJ opinion which currently prohibits filing charges against a sitting President) it is dubious that a President could effectively be held criminally accountable while in office.
After he leaves office, his only protection is relying on political concerns, or an anticipatory pardon.
Source? As far as I am able to tell, this is false; as in it is not that they are immune from prosecution, but that prosecution has never occurred.
Clearly, if war crimes were ever brought, regardless of the laws, commands, etc - it would have not resulted in being immune from prosecution.