The skills are:
The echo does not need (nor can it) climb or jump or swim, etc. It simply is moved 30 feet at a time by the knight. No check is required for this.
The echo is not granted the ability to lift or carry or drag objects so no check is able to be made.
When grappling or shoving, the rules say that it is the knight that is making the attack/grapple/shove check, not the echo.
As above, the echo does not need to attempt to maintain balance or swing from ropes, etc. It simply is moved 30 feet at a time by the knight. No check is required for this. In addition, it is immune to all conditions so it would never need to make an acrobatics check to escape a grapple.
Sleight of Hand
The echo is not granted the ability to pick pocket (a la Arcane Trickster Rogue's ability Mage Hand Legerdemain) or manipulate the environment in any way.
The echo, as above, is simply moved by the knight. It does not possess the ability to move of its own accord and so it can't move stealthily or attempt to hide. Moreover, since the knight is the creature making the attack, the rules for unseen attackers, etc apply with regard to the knight, not to the echo. This can get weird, depending on how you interpret the rules (this closed question is relevant).
The knight would be the one to make any of these checks because the knight has the knowledge and ability to reason, not the echo: Arcana, History, Investigation, Nature, Religion.
The knight would be the one to make any such check because the echo can't manipulate the environment.
The knight would be the one to make any such check because the echo can't reason.
The knight would be the one to make any such check because the echo can't reason and the knowledge would reside with the knight.
The knight would be the one to make any such check, even after it gets the feature Echo Avatar at level 7 because the rules clearly say "You [the knight] can see through your echo's eyes and hear through its ears."
The knight would be the one to make any such check. The echo can't track or reason, doesn't need food/water, etc.
The echo does not possess the ability to speak. It is possible to imagine a situation where the echo is used as a decoy of some sort (call it a passive deception). But as the echo is created by (and an image of) the knight, the onus would be on the knight to change its appearance to look like whatever it wanted the decoy to look like and therefore, as a DM, I would still require the check from the knight, not the image.
Similar to Deception but with the added caveat that, if a knight were to, say, try and intimidate a creature by making that creature think that it was outnumbered by more than one knight, I might rule that the echo's presence helped in some way but it would still be the knight that makes the check.
Similar to Intimidation or Deception.
Similar to all the other Charisma-based checks.
Given the limited capabilities the echo has (which is to say none that the knight doesn't take itself) I can't think of a situation where the echo would be required to or capable of making an ability check on its own.