How do I apply the distance-weighting matrix in a spatial autoregressive model
The weight matrix that you have illustrated needs to be row-normalized, i.e. the row-sum must be one in order for the rho parameter to have the proper parameter bounds, i.e. between (-1/lambda,1), where lambda is the smallest real eigenvalue of the W matrix.

When the weight matrix is row normalized and there is a change in the dependent variable, the shock in that other country will be an average of the surrounding values.

I'm not sure if this helps but a good resource is the LeSage and Pace text, Introduction to Spatial Econometrics, CRC Press 2009.

You can also visit www.spatial-econometrics.com to download free PDF books on spatial econometrics as well.