Cubby behind driver's seat
 

What's in the cubby behind the driver's seat. I know the passenger side has the fuel pump, but in pics I'm seeing something similar on the driver's side as well.

The main fuel level sensor unit, fuel filter, and fuel pump is on the right side (passenger side). The sub fuel level sensor is on the left side (driver's side).

Also, only the PS has a true cubby. The Z33 lacked a true glovebox, so Nissan decided a lined box was the best option on the PS. In Touring models, the BOSE subwoofer was placed behind the driver in '03-06 models.

Will the stock passenger lined box fit behind the driver side?

Yep - it's been done in the past. Just search glove compartment swap or glove box swap.

Top or bottom?

Top (small) is just a "glove box" as mentioned above.

Lower (large cover) houses subwoofer if you have Bose audio. (Has speaker perforation holes on cover). Non-Bose is just an empty compartment. Suitable for aftermarket amp and/or sub setup or stash, take your choice.