If you have a green card (and most likely you'd need to renounce your other citizenships) you can work there. ITAR and other CUI (and even a lot of classified stuff) is only restricted for foreign nationals.
Yes. And you are not a foreign national. ITAR does not apply to US citizens OR permanent residents. It even allows some refugees and asylees.
In fact, Clifford Chance LLP recently settled after improperly terminating several green card employees, believing that they couldn't work on ITAR projects[1].