Gandalf pursued the monster for eight days, until they climbed to the peak of Zirakzigil, where the Balrog was forced to turn and fight, its body erupting into new flame. Here they fought for two days and nights. In the end, the Balrog was defeated and cast down, breaking the mountainside where it fell "in ruin".
In addition to the twenty-eight named above there were also Gothmog, slain by Ecthelion, and another Balrog slain by Glorfindel... coming to a total of 30 plus however many slain by the men of Rog, however many slain during the fall of Thangorodrim, and the Moria Balrog and any other survivors (35 minimum).
Gandalf was able to kill the Balrog, but he never considered fighting Smaug. Instead, they sent Thorin and his company of dwarves to do the job while he dealt with the Necromancer in southern Mirkwood.
Glorfindel fought the Balrog and succeeded in slaying the monster, but was himself mortally wounded. His body was recovered by the great eagle Thorondor and buried under a stone cairn, where afterward grew yellow flowers.
The Elf Ecthelion leads the charge against the Orcs, and fights Gothmog, the greatest Balrog; they wound each other and both fall into the king's fountain in Gondolin; both drown.
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.
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. Their connection with the ancient foe known as Morgoth makes Balrogs far more fearsome than other enemies in the eyes of Elves.
The Battle of the Peak was Gandalf's name for his climactic battle with the Balrog known as Durin's Bane, on the peak of the mountain called Silvertine. Gandalf pursued the Balrog from the deepests dungeons of Moria to Durin's Tower, climbing through the whole Endless Stair.
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.
Secondly, Balrogs were the epitome of malice and power while Sauron only ever won one of three fights he ever had. Sauron was the great deceiver. His power lies in planning, subterfuge, manipulation, and corruption.
The Balrogs, or Valaraukar (singular Valarauko, Quenya for "Demon of power"), were three to seven Maiar seduced and corrupted by Morgoth into his service, turning into dreadful Úmaiar and scourges of fire cloaked in shadow.
Gothmog, one of the first Balrogs to become Morgoth's allies in the First Age, is regarded as the most powerful Balrog in Middle-earth. As the High Captain of Angband, he is given the title 'Lord of Balrogs'. Although he carries a fiery whip to battle just like the rest, he is also renowned for having a large black ax.
No, Balrogs do not serve Sauron. Similar to Sauron, Balrogs are Maiar, powerful spiritual beings second only to the Valar themselves. They were corrupted by Melkor's evil before the first age and were made into Valaraukar (Demons of Power). They very strength and wills were bound to Melkor.
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.
Although both the Balrog and Smaug are creatures of fire, and Gandalf is able to match them thanks to the power of his fire ring, defeating the Balrog takes far more spiritual prowess, which is why Gandalf is able to essentially 'level up' and comes back as Gandalf the White, whereas defeating such an immense physical ...
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.
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.
The combination of using both the actor and a CGI stand-in for certain scenes (such as his fight with Bolg) made the contrast starker, especially when compared to his more agile self in the Lord of the Rings films.
However, in the Tolkien novel, Gandalf actually never knew about the Balrog. All he knew was that Moria had been evacuated because of something called Durin's Bane, and that it still lurks inside. And while he does know that Orcs attacked the Dwarven kingdom, it was many years ago, and so he hoped they were gone.
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.
