Answer by ScottS for Tolkien's history of Elves singing and Ents walking and talking

Elves are Likely (Just) Older than Ents

Fourth option

This seems to be the correct sequence from my take on the information from the answer I accepted (see my comments below that answer also) along with the quotes given in the question and other answers here, but particularly how these things relate to the quote from Gandalf that sparked the question.

I felt a summary like within my question would be a fitting, useful addendum as an answer.

1. Trees were created (significant to the history of the Ents and Treebeard, see below*).
2. The second star creation occurred (significant to the history of the Elves).
3. Elves were created, but did not themselves yet talk, and in fact walked the earth under the stars without speaking for some time.
4. Ents were created (nearly simultaneously to Elves, but I think just after), as spirits summoned by Yavanna's awakened thought to inhabit some of the trees, and these Ents were walking from the start while also not speaking.
5. Elves began to talk and sing.
6. Elves taught (or awakened the desire in) Ents to talk.
7. The Huorns (likely) descend in some way from the lineage of the Ents after they could speak, for the Huorns also have a voice.

So the power that walked to which Gandalf refers is almost certainly the power of the Ents as Shepherds of the Trees, even before they were speaking.

That Ents are the oldest and most ancient of people (see my discussion under the accepted answer), my take is that the references are all in relation to the 3rd Age time frame. That is, Fangorn is the most ancient of any people group still surviving into the 3rd Age, and he belongs to the Ents. So through him, at the time of the 3rd Age, Ents are:

1. "oldest of living rational creatures" (emphasis added), i.e. the people group that still has a representative living as of the 3rd Age; I cannot imagine a non-living rational creature, nor do I think did Tolkien, and so that fact seems key to what is being stated in that quote from Letter 131 given in this answer.
2. "The most ancient people surviving in the Third Age" (emphasis added), i.e. the people group that still has a representative surviving as of the 3rd Age, which is Fangorn/Treebeard. The Elves had Círdan, and Tom Bombadil is not part of any "people" per se.

But also, see further thoughts related to "oldest" below.

* The quote from the Treebeard entry on Tolkien Gateway says:

Treebeard was the eldest person of Middle-earth, obviously being created along with the Ents during the Years of the Trees [me: but this is not so obvious, per the whole discussion here], before the creation of the stars; although he said that there were trees in Fangorn that were "older than he" [bold added].

The last statement actually fits with the time frame above and with the fact that spirits were summoned upon the awakening of the Elves to inhabit trees. So now we end up with some possible "overlapping" time frames for Treebeard even, namely:

1. Treebeard's tree "body" (hröa) likely existed before the Elves creation, then...
2. His "spirit" (fëa) joined the body, which occurred just after Elves awakening.

In this way, he is both "older" than Elves (in body), yet slightly younger than the Elves as a sentient combined body/spirit being (but as of the 3rd Age the oldest of any such species still surviving), and thus still younger than some of the trees in Fangorn forest that did not become Ents.