:goodpost: I agree with her thoughts.
I have used 2x2s with 1/4 plywood to make 1/2 sized doors and it works fine for that, but I have not done it for a full sized door.
I primarily use deck screws for all of my building that way I can disassemble if necessary. I vary the length of screw based upon what I am connecting. With 2x2s it is important to drill a pilot hole before using a deck screw or you can split the board. I also would use deck screws with OSB.