Cirdan was probably quite in touch with things related to the Straight Way. Recall, Gandalf disembarked right in his harbor. I don't think it was too hard for him to figure it out.
The Nazgul and the Mouth of Sauron seemed to know Gandalf, and I don't think his contending with all Nine could have been explained any other way.
I would think over the 2,000 years since their arrival that Sauron had idenified the five Istari and largely guessed at their origin. It no doubt troubled him, but he knew that they would not directly interfere any more than the Valar themselves would, and that they could be corrupted or distracted by worldly things and pursuits, as proved true for at least Saruman and Radagast.
What the Balrog knew? Who knows? He likely had no more than felt an unusual presence, until the encounter on the bridge. Gandalf's exclamation was essentially a calling card.
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The hoes unrecked in the fields were flung, __ and fallen ladders in the long grass lay __ of the lush orchards; every tree there turned __ its tangled head and eyed them secretly, __ and the ears listened of the nodding grasses; __ though noontide glowed on land and leaf, __ their limbs were chilled.
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