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.
Echo Knight is sloppy and is going to require some work from your DM
At this point, we can't avoid saying it. Usually, WotC writes very tightly and with great purpose. Echo Knight, however, has a lot of problems. If you want to play this class, make sure your DM is up for making a lot of rulings.
It seems clear from the language that the intention is for your reach to encompass the space around your echo, but the language is so imprecise and vague. The rules for reach are just kind of hinted at.
RAW reading - the grappled stays grappled
I'll try make a strict raw reading of the rules for you, but like I said before, this is going to be up to your DM.
The echo can be used to grapple: from Manifest Echo: "When you take the Attack action on your turn, any attack you make with that action can originate from your space or the echo's space.", from Grapple: "When you want to grab a creature or wrestle with it, you can use the Attack action to make a special melee attack, a grapple." So for starters, you can use the echo to grapple.
The next problem is the Grappled condition, as you note: "The condition also ends if an effect removes the grappled creature from the reach of the grappler or grappling effect" notice the use of the word "removes"?
Remove means: "take (something) away or off from the position occupied." or "abolish or get rid of." The question is, has the grappled creature been removed from the reach of the grappler or grappling effect? From my reading, no. The grappled creature was never within your reach, therefore it could not be "removed" from reach. Alternately, if Manifest Echo is a vehicle for a "grappling effect", then the grappled creature has remained within reach of the grappling effect.
Best Answer
The Echo does not provoke opportunity attacks.
You pretty much highlighted all the relevant text:
And
The echo is an image and not a creature, hence it does not provoke opportunity attacks.
By contrast, the spell simulacrum creates an illusory duplicate that is partially real, but in this instance the spell defines that image as a creature. The echo knight's Manifest Echo does not define it as a creature.