The Balrog killed Durin VI, the Dwarf-King of Khazad-dûm, whereafter it was called Durin's Bane by the Dwarves. Avarice, principally for mithril, drove the dwarves to go too deep and awaken the Balrog.
For more than five millennia, the Balrog hibernated in his deep hiding place at the roots of the mountains in Khazad-dûm. He remained undisturbed throughout the Second Age and most of the Third, before the mithril-miners of dwarf-King Durin VI awoke him in T.A. 1980.
Durin's Bane appears very briefly in the seventh episode of The Rings of Power Season One, when Durin III lets a leaf given him by the Elves fall into a deep abyss newly discovered in the search for mithril. The leaf seems to awaken the Balrog, who roars ferociously.
After the War of Wrath that ended the First Age, almost all the Balrogs were destroyed, “save some few that fled and hid themselves in caverns inaccessible at the roots of the earth,” according to Tolkien. So it may have simply found its way there.
After disinheriting his son, King Durin banished Elrond and ordered the new mine to be sealed up, but not before he cast a leaf from the tree in Lindon into the mine, where it floated down and awoke a Balrog that had been lying dormant beneath the mountain.
Awakening of Durin's Bane Balrog of Morgoth - The Rings of Power - Ep 7
What woke the Balrog in Moria?
The Balrog killed Durin VI, the Dwarf-King of Khazad-dûm, whereafter it was called Durin's Bane by the Dwarves. Avarice, principally for mithril, drove the dwarves to go too deep and awaken the Balrog. The Dwarves attempted to fight the Balrog, but its power was far too great for them.
So, when Gandalf agreed with Frodo's decision to travel through Moria, he knew that something could be hiding there. Yet, he had no reason to think that it would be something as awful and powerful as a Balrog. Only King Dain Ironfoot knew that Durin's Bane was a Balrog.
The fear of Elves is more spiritual, so the demonic creatures of shadow and flame are unsurprisingly horrifying to them. Legolas doesn't appear to have any hesitation in his actions throughout the films, but it's clear that his blood runs cold when a Balrog appears in the Mines of Moria.
While the colony was initially successful, Orcs and Durin's Bane, the Balrog, attacked them five years later, killing all of the Dwarves. However, Gimli did not know about the Dwarves' fate in The Lord of the Rings: The Fellowship of the Ring.
Gimli didn't know about Moria's fate because there had been no communication between Erebor and Moria for 25 years, and the Dwarves were hopeful and did not want to accept failure until there was certain news.
Gloin, also in The Hobbit, and his son, Gimli (a member of The Fellowship of the Ring), are also descendants of Durin IV, as a direct descendant as well, with Durin IV as his 9x great grandfather.
The only recorded instances of Balrogs having been slain are Gandalf's slay of Durin's Bane, Echtelion's slay of Gothmog and Glorfindel's slay of an unnamed Balrog.
In all of Tolkien's text I've read, there are three people who have killed a Balrog. Ecthellion, Glorfindel and of course Gandalf. All of these fights ended in a draw with all three sacrificing their lives for the deed.
Though Gimli has no children, he is not the last Dwarf of Middle-earth. After the fall of Sauron, his people continued to thrive in the Lonely Mountain. Gimli made his home in Rohan, in the Glittering Caves of Aglarond, and many of his kinsfolk came with him.
Long it stayed beneath the realm of Khazad-dûm, until the Dwarves there mined too deeply for mithril and unearthed the Balrog of Morgoth. The creature slew their king, Durin VI, earning the name "Durin's Bane", drove the Dwarves out and claimed Khazad-dûm (now called Moria) as its domain.
But I think you answered your own question. Gandalf doesn't know what it is until he sees it out in the open, as Legolas does as well. He recognized what it is as well as the elf, by sight, but is unable to recognize it just by confronting its power with his own.
As a Valar, Morgoth led a whole slew of Balrogs in the First Age, but because Sauron was only a Maia, the Balrog would have had no loyalty to him. He was his own master in Moria, and no Dark Lord was going to ruin that.
Sauron's deepest fear was that Aragorn would use the Ring against him. With its powers of domination, Aragorn could comfortably take over the entirety of Sauron's dark army. Though this would eventually corrupt Aragorn, it would allow him to destroy Sauron and take his place.
Sauron never becomes bound to Middle Earth like Melkor or the Balrogs during the first age because Sauron kept shifting shape, Sauron takes more different forms than any other being in the legendarium. Thus he did not become bound to a single form like Melkor and the Balrogs.
There had been no communication from Moria for quite some time, but no one knew exactly what had happened there, or why the colony had gone silent. Gandalf definitely did not know about the Balrog, or even about the orc attacks that had sent Balin and all who were with him to an early grave.
