Is there a 'Nyquist Criteria' for the minimum samples per chip that are needed for a correlator? I've heard 2, but will 1 work? What are the implications of having less than the minimum (in the context of radar and communications)?
Actually much less than 1 sample/chip will work for DSSS. Doing so erodes the spreading gain and makes chip level detection impractical, but it works. The undersampled spreading sequence simply acts as a vector (subspace) for the undersampled signal. Matched filtering is still matched filtering -- as long as you are still sampled well above the symbol (not chip) rate.