HID descriptor to use for a Playstation 2 controller
You can't have pressure sensitive buttons. You need to specify them as additional axes.

Additionally, the controller really has 12 pressure sensitive buttons? In that case, you're out of luck, as I have yet to find any software that works with more then 8 axes (including windows!).

Furthermore, you can consolidate some of your axes controlls:

const PROGMEM USB_Descriptor_HIDReport_Datatype_t hidDescriptor[] =
{
    0x05, 0x01,                    // USAGE_PAGE (Generic Desktop)
    0x09, 0x05,                    // USAGE (Game Pad)
    0xa1, 0x01,                    // COLLECTION (Application)
    0x09, 0x01,                    //   USAGE (Pointer)
    0xa1, 0x00,                    //   COLLECTION (Physical)

 // ----- Consolidated
                                   //     ; Right stick
    0x09, 0x33,                    //     USAGE (Rx)
    0x09, 0x34,                    //     USAGE (Ry)
                                   //     ; Left stick
    0x09, 0x30,                    //     USAGE (X)
    0x09, 0x31,                    //     USAGE (Y)
    0x15, 0x00,                    //     LOGICAL_MINIMUM (0)
    0x26, 0xff, 0x00,              //     LOGICAL_MAXIMUM (255)
    0x75, 0x08,                    //     REPORT_SIZE (8)
    0x95, 0x02,                    //     REPORT_COUNT (4)    //changed
    0x81, 0x02,                    //     INPUT (Data,Var,Abs)

    0xc0,                          //   END_COLLECTION


    0x05, 0x09,                    //   USAGE_PAGE (Button)
// this section isn't going to work...
                                   //   ; Pressure sensitive buttons
    0x19, 0x01,                    //   USAGE_MINIMUM (Button 1)
    0x29, 0x0c,                    //   USAGE_MAXIMUM (Button 12)
    0x15, 0x00,                    //   LOGICAL_MINIMUM (0)
    0x26, 0xff, 0x00,              //   LOGICAL_MAXIMUM (255)
    0x75, 0x08,                    //   REPORT_SIZE (8)
    0x95, 0x0c,                    //   REPORT_COUNT (12)
    0x81, 0x02,                    //   INPUT (Data,Var,Abs)
// this section will, though the previous section will likely break something, and the whole descriptor will likely be rejected.
                                   //   ; Other buttons (L3, R3, Select, Start)
    0x19, 0x0d,                    //   USAGE_MINIMUM (Button 13)
    0x29, 0x10,                    //   USAGE_MAXIMUM (Button 16)
    0x15, 0x00,                    //   LOGICAL_MINIMUM (0)
    0x25, 0x01,                    //   LOGICAL_MAXIMUM (1)
    0x75, 0x01,                    //   REPORT_SIZE (1)
    0x95, 0x04,                    //   REPORT_COUNT (4)
    0x81, 0x02,                    //   INPUT (Data,Var,Abs)
    0xc0                           // END_COLLECTION
};


What I suspect is happening is the computer is interpreting a 0 for the value on the buttons as "not pressed", and anything else as "pressed". As such, unless the button completely returns to the unpressed state, the button will seem to "stick".

Anyways, the generic desktop usage page of the HID usage table actually only defines a total of 9 analog axes in it's entirety, so you're kind of out of luck. Even the windows controller diagnostic only supports a maximum of 8 axes, and most libraries that access gamepads support even fewer. 

DirectX, for example, only allows you to access 6 analog axes.

Anyways, take a look at the USB HID usage tables here.

I also spent a while playing with HID tables in the process of converting an arduino leonardo into a joystick. Look at that here. 

Incidentally, if you can, I recommend a Arduino Leonardo for this sort of tinkering. It's a lot easier to deal with then trying to re-flash the ATmega8U2 in a Arduino Uno.