Christofides algorithm (by hand) (suboptimal solution - is it my fault?)
As mentioned by Yuval, Christofides’ algorithm is an approximation algorithm to the travelling salesman problem. It is not guaranteed to produce an optimal solution. So it is not unexpected that you could end up with a sub-optimal solution of 

On the other hand, you did make a mistake while computing the minimal spanning tree. In your step 1 that calculates the minimum spanning tree, edge H$\alpha$ should be replaced by edge HG.