Can the imperial officer order the e-web engineer to move and the engineer can attack later in his own activation phase?

Or is the movement order to consider as voluntarily leaving it's space which prohibits to attack?

And can the engineer move and perform an attack after receiving the order from an elite officer?

The E-Web can shoot on its own activation if it was moved by an officer or something else earlier.

During your activation, you [the E-web] cannot voluntarily exit your space if you attack, and you cannot attack if you exit your space.

The restriction only applies "during your activation." In this case "your" is the activation of the E-web. If an Officer tells it to move or shoot, it's still the officer's activation, so the E-web restriction does not apply.