Model validation error and test error Does choosing a model with the lowest validation error also guarantee that the model has the lowest test error?

No.
First, there is always residual variance, so even if the model with the lowest validation error has the lowest expected prediction error, for any given dataset, it may still have a higher error than a competing model.
Second, there is such a thing as "overfitting to the validation set": if you fit and tweak your model, then assess it on the validation set, then go back and tweak some more, iterating the process, then you will end up with a model whose predictions closely follow the validation set - and you may overfit here. Essentially, once you go back to the modeling stage after looking at the validation set, it is not a validation set any more. And once you have overfitted to your validation set, your performance on a true holdout test set may of course be worse than for a non-overfitted model.
