As per my comment, you have specified UNO board, but it doesn't have Keyboard/Joystick required definitions for the board (I'm not sure if UNO has any buttons to be defined there, it only has Leds, and leds are defined in Led.h - check out \LUFA-111009\LUFA\Drivers\Board\AVR8\UNO)
So what you could do is to create Board folder under your KeyboardMouse and make empty files Joystick.h and Buttons.h there. This should get you going further. Errors you are seeing are due to the following code in \LUFA\Drivers\Board\Buttons.h
#if (BOARD == BOARD_NONE)
#error The Board Buttons driver cannot be used if the makefile BOARD option is not set.
#elif (BOARD == BOARD_USBKEY)
#include "AVR8/USBKEY/Buttons.h"
....
#else
#include "Board/Buttons.h" <------ THIS IS EXECUTED SINCE UNO DOES NOT HAVE BUTTONS
#endif
So your error
../../../../LUFA/Drivers/Board/Joystick.h:119:31: error:
Board/Joystick.h: No such file or directory
means that your folder structure and your LUFA configuration is correct, but you're missing file Buttons.h in your KeyboardMouse/Board/ folder. Got it?
Try what I've suggested and see how far you get. You can see how to define buttons in other Board's folders, for example in LUFA\Drivers\Board\AVR8\USBKEY\
EDIT
Btw, I forgot to mention, error about common.h should go away hopefully after fixing this since that ........\ is coming from a file in a different location in folder structure thus confusion.
EDIT
OK, so here's the link on how to build custom board drivers:
http://www.fourwalledcubicle.com/files/LUFA/Doc/111009/html/page_writing_board_drivers.html
What you need to do is to copy files Buttons.h and Joystick.h LUFA\CodeTemplates\DriverStubs\ (or try copying Buttons.h and Joystick.h from USBKEY better, I think you still would need to specify a value for each definition otherwise) This should get rid of undefined errors. You have TODO sections in the files that you need to update.
OK, so I think I should also mention how this is supposed to be used before going further. These drivers/definitions are meant to be used in a specific manner in your code, and LUFA is unifying the approach for you. As far as I can tell, buttons are used in a following manner:
if (Buttons_GetStatus() & BUTTONS_BUTTON1){ ... do something when button 1 pressed....
This way, if you have several boards with at least one button, your code should theoretically stay the same across the boards.
Similar stands for LEDs, you can use them like:
LEDs_SetAllLEDs(LEDMASK_USB_NOTREADY);
....
LEDs_SetAllLEDs(LEDMASK_USB_ENUMERATING);
I hope you get the picture. In order to use these library functions you have to define buttons/joystick/led specifics in their respective header files. So for example - in buttons.h you need to specify any custom header files you need, add port masks for buttons (on which pin of which port they are connected), specify port initialization and how to read the status of the buttons. You can find all of that in the USBKEY's buttons.h - e.g. it's importing common.h, defines BUTTONS_BUTTON1 like pin 2 of a port, initializes PortE with this button (so button is pin 2 on port E), and in Buttons_GetStatus it reads the status of the button.
I could go on and on in the same manner for joystick as well, but I hope you get the picture. Joystick is more involved but it's like having 4 buttons of which 0, 1 or 2 can be active at any time.
BTW, this is only useful if you have any buttons on your board. For example, I made keyboard driver without any buttons (I had to remove buttons specific code though). I used Ir Diode to read remote control codes and make the board act as keyboard. So you don't really need the buttons, nor the joystick (of course, it completely depends on what you're doing).
Best Answer
Ok, so you need a device with at least 2 USB ports. One of these must be a slave port which you will connect to a host PC and the other port must be a host which you will connect a keyboard/mouse/whatever. Your device will then enumerate and handle the slave device(s) on its host port, process any information and/or events from these devices, and then forward this data out of its slave port up to the host PC. So to answer your question simply: yes this is possible.
But, I don't think its going to be possible with either of the boards you're looking at. The one half would be fairly straightforward. These boards would be quite happy using one of their USB ports to talk to your keyboard/mouse device(s). The problem comes about when you want to use them to emulate a particular type of slave device to your host PC. These kind of boards are typically configured to provide a very limited selection of USB profiles to choose from when connecting them to a PC (such as a mass-storage device, serial port emulation, camera or possibly some kind of debugging interface) and you'd find it quite difficult to change this to look like something else - in your case a HID device like a keyboard and/or mouse. I'm not saying its impossible, but I don't know enough about these boards and its not obvious from their simple overviews that they support this.
This is the kind of application where you will probably need to work at a lower level - more directly with the firmware on a suitable microcontroller or microprocessor.
If I were to try to pull something like this off, I'd be inclined to set up a pair of microcontrollers back-to-back. One of them is the USB OTG host to your keyboard/mouse, the other one acts as a keyboard/mouse compound device to your host PC, and they're connected to each other using SPI.