My response is quite late but since nobody has answered this question let me give a try. Maybe it'll benefit other people with similar doubt.
Check whether the uncontrollable states are stable using the Eigen vlaues. There is a difference between controllability and Stability. Its quite possible that the states which are not controllable are already stable. In that case you can design LQR controller to stabilize and control the states which are contollable.
Regarding selection of Q and R, there is no specific rule here. Its just trial and error. You have to start with a nominal guess (I suggest use Bryson's rule here) and then see your system's time response. Retune the matrices to get the desired response.
Hope this helps!!