The knight gains the ability to make an extra attack, not the echoes.

The echo knight is who is granted the extra attack by Unleash Incarnation and the feature says this extra attack is granted, if the knight decides to use the feature, when she takes the Attack action.

The feature also allows for the knight to choose to make that additional attack (and not the knight's normal attacks) from the echo's location.

The important point here is that the echo itself is never granted an attack.

When the Unleash Incarnation feature is later improved by the Legion of One feature, the echo knight still only has the one extra attack but, owing to the fact that Legion of One allows the knight to have two echoes simultaneously, the knight can choose which echo's position to make that attack from.

The knight can make additional extra attacks from Unleash Incarnation only if it can take the Attack action additional times. The obvious way of doing this is through the fighter's Action Surge ability. Crucially, the same limitation applies: each additional time the Attack action is taken, only one extra attack is granted.

Quick side note: Haste would not grant the opportunity to use Unleash Incarnation because the spell explicitly limits the attack action it can grant to a single attack. See Does the Echo Knight fighter's Unleash Incarnation feature add another attack to the additional Attack action from the Haste spell? for more info.

To spell it out more directly, a fighter of 18th level would typically have 3 attacks per Attack action. Using Unleash Incarnation with the Attack action would give the fighter a 4th attack. If the knight Action Surges, it could do that again for 8 total attacks, excluding any potential attack granted by its bonus action.

Note that if the knight did have an attack available through a bonus action (such as from the Great Weapon Master or Polearm Master feats), this would not allow the knight to use Unleash Incarnation yet again, because that bonus attack is not granted by "taking the Attack action", which is the necessary prerequisite for getting Unleash Incarnation's extra attack.

No, it cannot (usually).

The class feature description of the echo knight's Manifest Echo gives no guidance here, but the echo knight subclass description is not all features - Explorer's Guide to Wildemount (p. 183) motivates for us exactly what an echo knight's Manifest Echo ability is doing:

A mysterious and feared frontline warrior of the Kryn Dynasty, the Echo Knight has mastered the art of using dunamis to summon the fading shades of unrealized timelines to aid them in battle. Surrounded by echoes of their own might, they charge into the fray as a cycling swarm of shadows and strikes.

But Thomas, that's just flavor text! Sometimes flavor text is not just flavor text, particularly when a natural application of the flavor text very clearly motivates how a particular ability functions.

In this case, I've bolded the key phrase; your manifested echoes are "fading shades of unrealized timelines". Your echo is an alternate timeline of yourself, which naturally means that the echo cannot do anything you cannot do, unless otherwise specified by the various subclass feature descriptions. If there is no alternate timeline in which I can move vertically or pass through walls, then there is no vertically moving or wall-pass-throughing echo to manifest.

But I have a flying speed!

Then your echo has a fly speed, maybe. If flying (read: moving vertically) is innate to your character, such as an Aarakocra echo knight, then alternate timelines where you move vertically are totally feasible, and your manifested echo could be one of them.

But what if I can cast fly? I'm going to call this DM purview, but I personally lean toward "Sorry, you cannot choose to manifest the echo that had just cast fly". Nothing in the feature descriptions allow an echo to generally benefit from buffs you have or have available to you.

But I can normally pass through walls!

Can you, though? Per PHB, p. 7:

an adventurer can’t normally pass through walls

I'm not aware of any racial or class features that just allow you to pass through walls willy-nilly. As for spells or magic items requiring activation, the reasoning I outlined concerning use of the spell fly applies.

As always, the DM may rule otherwise.

After receiving some feedback in chat, user GcL helpfully pointed out:

I think the presence of a floor in one timeline does not constrain the presence nor location of a floor in another timeline.

Following the logic of different timelines, the [echo] "in the air" is actually standing on a floor or ground that does exist in the timeline they're drawn from.

This is not an entirely unreasonable argument, so it would not be entirely unreasonable for a DM to permit the echo to move vertically or pass through walls, especially given the apparent ambiguity of the feature description.