Q1: Why do think Thingol had troubled dreams of Men, even before he had heard tidings of them. What kind of 'troubles' do you think he envisions? How do you think he came to know of the Secondborn? Melian? Rumor from other Elven lands? The Noldorin?

I will say this right now: I do not like the portrayal of Thingol in the Silmarillion. Most of the Elves in the First Age behave better in the Silmarillion (and better yet in LotR) than they do in previous versions of the story (e.g. BoLT). Thingol is the exception - in BoLT, Tinwelint (so Thingol is named in BoLT) is a much more likable character than he is in the Sil. This is another reason why I don't think Tolkien ever published the Sil: how could a total S.O.B. like Thingol ever be somebody to whom Melian would find attractive, and how could he ever sire such a wonderful person as Lùthien?

Q2: Do you think Melian refers only to Beren? Or also Hurin ?

As for our Adan, it has to be Beren. Just as Lùthien rises out of obscurity to rival and supass Fëanor, Beren does the same to all of the great warriors of Tolkien's legendarium.

So how does Tùrin get into Doriath? I don't think that he's as unwelcome as Beren was at first, since Beren did, after all, succeed in that spectacular quest. At least Thingol learnt that lesson - too bad he didn't learn about slighting Dwarves.

Q3: Do you think the ‘doom’ is the parting with Luthien, the coming of the Silmarils, the Fall of Doriath, or all of these? Other thoughts?

It's the Fate of Arda. The Silmarils contain the hope of the Free, even a substantial amount of the power of Varda. As long as Melkor has all of them, He cannot be overthrown, not even by the Valar.

How to save the world? Enter the perfect heroine and the perfect hero, who, as a result of an incredible quest combining courage, power, and epic heroism, to make this one hole in Melkor's invincible armor... and when the time comes, the Valar will defeat him.

The Will of Ilùvitar is at work!