When Morgoth's defeat was inevitable, the Balrog fled into the tunnels deep within Middle-earth. They were created by the infamous Nameless Things, and they offered a surefire place to stay hidden. And it did, until the Dwarves delved too deep during the Third Age, disturbing its slumber.
During these ancient times, the Balrogs were led by Gothmog, but when Morgoth was defeated in the final year of the First Age, they retreated into the depths again, and this is how Durin's Bane came to be in Moria. The total number of Balrogs in Tolkien mythology is somewhat contentious.
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.
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.
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.
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.
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.
The company fled with Gandalf, but the Orcs and the Balrog, taking a different route, caught up with them at the Bridge of Khazad-dûm. At the Bridge, Legolas instantly recognized it as a Balrog, and Gimli recognized it as Durin's Bane.
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 name “Durin's Bane” literally means “that which destroyed Durin”, so it's fairly certain that the name referred only to the balrog in Moria. The dwarves didn't know what it was, since none of them had ever actually seen it. They just knew something they had never encountered before had killed Durin.
Gandalf's main reason for supporting Thorin's quest in The Hobbit was to eliminate the threat of the dragon Smaug, who would be a powerful ally for Sauron. Despite popular belief, the Balrog known as Durin's Bane was more powerful than Smaug in Middle-earth, being a literal, primordial demon.
In the end Tolkien stated that there were probably "at most" seven Balrogs: In the margin my father wrote: 'There should not be supposed more than say 3 or at most 7 ever existed. '
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.
The Balrog, like Gandalf, is a Maia. There is a very real possibility that Gandalf will die in this confrontation. As an istar Gandalf is an incarnated maia, and thus subject to all the frailties and fears that Elves and Men are subject to.
Smaug could easily have cut through the armies of Mordor, and delayed the invasion of Middle-earth by years. So it wasn't in Sauron's interest to take on the dragon, but that doesn't mean he saw it as a lost cause, as he fully planned on creating an alliance.
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.
Gimli was born in the Ered Luin in the Third Age, son of Glóin. Gimli had wanted to accompany his father on the quest to reclaim Erebor, but at age 62 he was deemed too young. He was a remote descendant of Durin the Deathless, chief of the Seven Fathers of the Dwarves and ancestor of the Longbeards.
Balin was distantly related to him, and a descendant of the noble House of Durin. He later went to re-establish the Dwarf presence in Moria, but within five years it was overcome once more by Orcs, and he was killed there along with the rest of his companions.
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.
In general Sauron did not seem to be afraid of the Balrog during the Third Age. The behavior of Durin's Bane indicated that he would not leave Moria and its tunnels : he was not a rival for the control of Middle-earth and the Balrog did not share Sauron's abilities as a potential Dark lord.
