You've described the process perfectly Colin.
2 - 3 mm would be fine. Using wood glue and leaving overnight would be best.
After drilling the holes in the corner I would suggest using a fine blade for cutting, it may take you longer but it will reduce the chance of any tear-out and leave smoother edges.
I personally think it's better to cut the shorter sides first so that all your cross crain cutting is done before the wood starts to become more flexible.
Take it slow and remember the golden rule ... Measure Twice, Cut Once !
All the best,