The full set of rules are here: "Expanding on past practice, satellites of minor planets will, where possible and appropriate, receive names closely related to the name of the primary and suggesting the relative sizes. For example, binary transneptunian objects of comparable size should receive the names of twins or siblings, consistent with the current principle of using names of gods of creation or the underworld."

If there are not already objects named Nergal, Ereškigal, Namtar, there totally should be.
Not yet. Nergal is a little too closely associated with Mars, but the other two would be fine.
Though the trans-Neptunian objects have been going with Inuit (Sedna), Hawaiian (Haumea, Namaka, Hi'iaka), Rapa Nui (Makemake), Native American (Quaoar, Teharonhiawako) and Vedic (Varuna) mythologies in the last wee while. Not doing too bad ;)