Can you guess what G. was really after, hanging around Melian?     Virtual beer to first correct answer.

It has been kindly pointed out that Galadriel is in love with Celeborn. There's more, though, and that is that Melian's daughter is also an absolute virtuosa in the Elvish Arts. Lùthien does not apparently have the same protective and healing power that Melian does, but she has some. Moreover, her particular powers can be used as a weapon, indeed the most powerful weapon that the Elves ever have. As we learn in the appendicies of LotR, Galadriel apparently uses some of the same powers, although not on the same scale as the amazing stunts that Lùthien pulls off. I expect that Galadriel and Lùthien are very close friends, too!

Funny that in 300 years G. never gossiped with the girls about the trip over from Valinor.  Amazing restraint? Stiff-necked Noldor? Worried about Thingol's reaction if the truth got out?

300 years or 3,000? :)

I think that Thingol finds out eventually - certainly he does in the earlier versions of the Sil, and he forbids the speaking of Quenya within his halls as a disgrace to the Noldor.

Melian is puzzled, but smart enough to detect the curious lack of information about the trip.  G. is clearly now protecting her kin, evidently considers her loyalty to the Noldor more important than being up-front with an ally.  Or is she reluctant to lose Melian's friendship? What is going on here?

In UT, Galadriel fights for the Teleri, not against them. Nonetheless, I think she is wise to try to avert further kinslaying as the Sindar take their vengence.

Despite the Valar?  is that all? What about her wishing to rule her own realm?
Doesn't covering for Fëanor implicate her?   
Now Melian is aware that there is lots more to the story, and is impelled to seek out more information.

She's covering the monster, but she's covering innocents, too. I'm sure she feels somewhat guilty for it herself, though. I don't know.