Due to new information gathered from my question Does Rope Trick create an extradimensional space, or does the space already exist?, a lot of this answer falls apart. I am going to leave the answer as-is, since it has received a few positive votes. I still believe that the wording implies that the cells exist, simply because it never says that the cells are created. However, it does not explicitly state either way, so it may be up to the DM to make the decision.
The cells continue to exist, so there is nothing preventing Gate from stepping into the cell.
There are a couple of questions to address here.
"Can the pocket dimension created by the mirror of life trapping be returned to by using the Gate spell?"
The Mirror of Life Trapping simply states that
A creature trapped within a cell can escape using magic that permits Planar Travel.
It says nothing about re-entering. This, like you deducted, would probably depend on whether the extradimensional cell exists outside of being occupied. If the space exists, than I see no reason why the Gate spell couldn't be used to enter it.
"Does the extra dimensional cell remain in existence when the wizard leaves it, or collapse until the cell is refilled? What if the wizard leaves his notes, are they ejected when the wizard leaves?"
The wording of Mirror of Life Trapping implies that they continue to exist outside of being occupied:
Any creature other than you that sees its reflection in the activated mirror ... [will] be trapped, along with anything it is wearing or carrying, in one of the mirror's twelve extradimensional cells
If the cells did not already exist, it would probably read something like "An extradimensional cell is created, up to twelve."
If we compare it to the wording of Rope Trick, which also uses the term "extradimensional"
...an Invisible entrance opens to an extradimensional space that lasts until the spell ends.
This explicitly states that the space is created brand-new and is completely destroyed once the spell ends. Since such text does not exist for the Mirror, it's safe to say that they exist, anything left in them (after Gateing out) remains in them, and you can Gate back in at any time.