Who was aware that a Balrog lived in Moria?
Sauron almost certainly knew of the Balrog, at least through his Orcs and very possibly more directly. The Dwarves knew that “Durin’s Bane” was still in Moria when Dain saw it inside the gate at the battle of Azanulbizar, but they may not have known what it was: at the Council of Elrond, Gloin calls it simply “the nameless fear.” In “Lothlorien”, Celeborn tells the Fellowship, “We long have feared that under Caradhras a terror slept.” This indicates that he wasn’t sure anything was there, and suggests that he did not know the nature of the “terror”. Similarly, in “The Bridge of Khazad-dum”, Gandalf clearly does not know what to expect: after confronting the Balrog through the door of the Chamber of Mazarbul, he says, “what it was I cannot guess”. When the company finally sees it, he says, “A Balrog. Now I understand.” If neither Gandalf nor Celeborn knew of its presence, it seems unlikely that any of the White Council did.