We don't know WHEN any of those things happen. It looks like he's confronted by Celebrimbor in some ruins so presumably that's closer to the end of the season when things have already started falling apart. We also don't know that Galadriel's voice over is referring to Annatar specifically, just Sauron in general who they (as of the start of the season) believe has left Eregion.
As Tolkien wrote it Annatar was a sketchy elf that claimed he was an emissary from the Valar and even though some of the highest ranking elves distrusted him and refused to deal with him he was still fully accepted by the smiths. The theme is that hubris and a desire for power blinded at least SOME of the elves to the true nature of who was guiding them, but Tolkien didn't really flesh any of this out at all.
In terms of the show, using the same actor in order to maintain visual consistency makes sense, and until we know who exactly Annatar deals with directly there's no reason to assume his face alone is a giveaway within the context that it's presented.
As far as her and Elrond are aware Sauron has left Eregion so there's no real reason for them to remain there instead of going back to Lindon, or Kazad-dum, or perhaps even pursuing Halbrand back to wherever they think he might have gone. Celebrimbor having dealt directly with Halbrand is really the only "issue" but again we go back to the smiths of Eregion being blinded by the gifts that Annatar brought. What exactly those are we don't know. Within the context of the show The Three are unique because they were crafted with the Valinor metal, so maybe he brings something else to the table for the creation of magic rings that convinces them he's working in their favor.