Well, at least 2 axes should be pretty easy: the X and Y pots are just sitting in the black plastic brackets which have just been popped into the white plastic. You don't even need to de-solder anything to use those two pots so long as the wire is long enough. (Edit: actually,
Logitech hasn't even soldered the pots: they've just connected white plastic headers onto the posts of the potentiometers- I presume that the wires from the headers are soldered onto the controller board. That said, I imagine at least some of the switches- little micro buttons- are soldered directly onto the controller board- those may be trickier)
As I suspected, the pots won't have much travel (an arc of maybe 60-90 degrees).
You can either live with it as-is and mount the pots vertically in the existing little black brackets but you'll certainly have to extend the knobbly black bit with a handle of some sort as it is way too short right now to give you any precision for trimming. Or, in the alternative you can get some gears to make a more realistic multi-turn trim wheel.
For a first project you might want to go with the first option but it's your call