Given that linear discriminant analysis (LDA) for two groups and multiple regression essentially the same results, could they be used as confirmatory techniques? For example, could a DA be used to classify students in high vs. low performing groups, followed by linear regression to confirm DA predictions or vice versa? Would they always yield the same results or could the predictors differ?
No, it does not make sense to use one to confirm the other. That they give you the same answer is because they are doing basically the same thing. If one is bad, then the other will also be bad.
Sometimes they will have different predictors, but this is only due to different conventions (e.g., most people select variables for regression using significance tests, whereas with LDA cross-validation is more popular).
You can improve predictions by using multiple models and taking their average prediction (provided all the models are sensible), however, combing the results of LDA and linear regression would be pretty pointless as they are too similar (unless you used different predictor variables).