[Lord Gothmog]

To start with we have some points from the Silmarillion.

Now men awoke and listened to Felagund as he harped and sang, and each thought that he was in some fair dream, until he saw that his fellows were awake also beside him; but they did not speak or stir while Felagund still played, because of the beauty of the music and the wonder of the song. Wisdom was in the words of the elven-King, and the harts grew wiser that hearkened to him; for the things of which he sang, of the making of Arda, and the bliss of Aman beyond the shadows of the Sea, came as clear visions before their eyes, and his Elvish speech was interpreted in each mind according to its measure.

from, Chapter 17: of The Coming of Men into the West.

And

Now the Eldar were beyond all other peoples skilled in tongues; and Felagund discovered also that he could read in the minds of Men such thoughts as they wished to reveal in speech, so that their words were easily interpreted.

Same Chapter.

So the Elves seem to have shown this ability from the start. It would probably vary acording to the power and skill of each Elf.

As for the points raised in LotR

1. The best known is probably the incident at Amon Hen when Frodo heard the voice of the absent Gandalf warning him to remove the Ring.

Gandalf was a Maia and at that point had returned more powerful in Middle-earth than he had been.

2. Galadriel's mental probe of each Fellowship member:

Galadriel was a powerful Noldor who had come from Aman to Middle-earth so probably had a great deal of skill.

3. Gandalf's explains to Frodo who is waking at Rivendell how he already knows certain things about the journey which Frodo had not even shared with the other hobbits:

Same answer as above.

4. Pippin's images as he is driven by the Orcs down the trail:

This could have two answers. One is that either Aragorn or Legolas projected the image. Two is that there was no image projected but that it was a combonation of Pippin's memory of Aragorn and his Imagination giving him the idea that Aragorn would be following. He would not know of what happened after they were captured so it is only natural that he might think that the fellowship would follow them.

5. And, most poignant and striking, in "Many Partings":

Again except for Celeborn and Elrond (who was trusted to hold one of the Elven rings so must have been quite powerful) the others have already been answered for. Celeborn lived in Doriath at the time when Thingol and Melian were King and Queen, so was among the fairest and most powerful of the Moriquendi.

I hope that this is of some use to you.
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