Narya vs. the Valarauko
 

When Olórin defeated the Valarauko in Moria, did he dare to use Narya to combat the demon? Could he defeat his enemy without it? If he did use Narya, would Gorthaur have detected it? Did Gorthaur already know about the Valarauko in Moria?

Do Balrogs have wings? (j/k)

Quite why you feel you had to use old Quenya and Sindar names instead of the more traditional names used in the LoTRs I dont know. Anyway Gandolf (or Ólorin whatever) could not have used his ring to combat the demon as the Three Rings of the Elves could only be used for lore and knowledge, and gave the wearer

Sauron does not know where the rings are without the ruling ring, as it is said that was he to find the One Ring Lórien would be exposed, so while Frodo had the ring Gandolf could use the extra knowledge given by Narya without fear of exposure.

well,,,i don't know if sauron would detect gandalf or not since sauron didn't have his One ring. i think whether sauron did know bout the balrog or not i really doubt that he would try to bring it under his power. balrogs and dragons were melkor's most powerful servants(beside sauron himself). and sauron was more of a nazgul and troll kinda guy. don't even start bout "did balrogs have wings"!

Gandalf didn't use anything, he's just super-amazing by himself. The Elf Rings were to preserve stuff, not for fighting.