[Legate of Amon Lanc]

For me, it is completely out of question that it would be any of the "interested characters" sending the dream. By "interested characters", I mean basically anybody in Middle-Earth, and even less those who are part of the story, which effectively dismisses all Gandalfs or Galadriels...

Firstly, it would really make no sense that nobody ever mentions it. I mean, Gandalf or Galadriel would surely say, certainly at some point, "Yes, it was me who actually sent the dream", or it would be written in some of Tolkien's notes (I can imagine all these stories in UT with the behind-the-scenes stuff telling something about this). Gandalf himself was sitting at the Council when the question about the dream was brought forth. And (and that is the second point), wouldn't it be totally silly if Gandalf at that point had to say "um, yes, I actually know that dream, I have sent it myself?" Relatedly, I see no reason why Gandalf, or even less Galadriel, would send a prophetic dream to some random guy in Gondor. At least Galadriel: that would be indeed totally random, as Galadriel had nothing to do with Gondor. Gandalf would be more logical, as Faramir was his friend, but again, why then did also Boromir get the dream once? Perhaps only if Gandalf once "called the wrong number". In any case, once again, there is no indication that Gandalf knew anything about the subject before Boromir brought it up.

I think it could not have been anybody in Middle-Earth, because first you need to have a person who knows a) Faramir, b) Boromir, c) Isildur's Bane, d) Halflings, e) Sword that was broken, f) where it is. There really are not too many people like this. And most importantly, I don't like the idea because it would be outright manipulation, if somebody who is sort of personally involved in the matter would send a dream like this to Faramir (resp. Boromir).

No, in my opinion, it has to come from "outside the Middle-Earth": and at this point, it ceases to interest me, actually. "Something else at work" is always and has always been the best description for me. If somebody really pushed me, I would say either "the Valar" (as a whole), or "Eru" (which is sort of logical, but somehow I wonder if it is not unnecessarily high). The idea about Ulmo mentioned above is not relevant in my opinion, not anymore in Third Age; Ulmo was the last of the Valar to communicate with the exiles in First Age, but since then, many many things have changed and Valinor is outside the Circles of the World for good, and already by the end of FA Ulmo's power in the waters of Middle-Earth was waning. This was certainly a thing of importance, one of the few instances where there was "something else at work" in the fate of Middle-Earth, something from the outside, and in that case I believe it would have to be either just Eru himself or all of the Valar acting here together - just like with sending the Istari. So that is my opinion.