As for lebanon and hizbullah, the US stance is to negotiate and deal only with non-terrorist aligned members of the lebanese government. As for those affiliated with hizbullah and any other recognized terrorist group the US treats them as if they dont exist. This model has been floated as an idea for continued US involvement in Palestinian affairs after Hamas joins the PA.
Iraq prepares troops for offensive to retake Mosul (Reuters)
from the article: U.S. tells Israel it will shun PA gov't that includes Hamas