I am curious what it means exactly that PMF forces (including Kataib Hezbollah) are
designated [...] as an "independent military formation" within the Iraqi armed forces.
Do they actually get any funding from the government of Iraq (as opposed to just Iran, which allegedly funds them [too])?
A February 2024 article from Reuters about Kataib Hezbollah (KH) and the Popular Mobilization Forces (PMF) says KH "fighters receive state salaries".
Kataib Hezbollah is a Shi'ite group which had a long history of violence against US forces in Iraq but was included in the Popular Mobilization Forces, which were set up by the government at in 2014 to combat Islamic State. PMF was set up at a time of crisis includes various militas who could be relied upon to be hostile to IS even if they had also fought the government or were linked to Iraq.
KH is described as "shadowy" and its precise income and funding probably isn't known by anyone, although it's reasonable to suspect Iranian involvement. It operates both as part of the security forces and as a law unto itself. Such is the situation in Iraq.
Kata'ib Hezbollah has received extensive training, funding, logistic support, weapons, and intelligence from the Quds Force of Iran's Islamic Revolutionary Guard Corps (IRGC).[46]
https://en.wikipedia.org/wiki/Kata%27ib_Hezbollah
No, it doesn't seem like the U.S. believes they are funded by Iraq. It seems they receive funding from Iran. The Hezbollah Brigades is a radical Iraqi Shiite paramilitary group, and should not be confused with Lebanon's Hezbollah, which provided weapons to them. Lebanon's Hezbollah has been purported to be receiving funding from Iran, Iraq, Syria and Qatar.
