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Big-O notation is a theoretical tool, it's somewhat useful in practice but it won't necessarily tell you which of two algorithms is faster. It doesn't tell you about cache performance, memory requirements, or even if there's a large coefficient on that n * log(n) term.

That's for comparison-based sorting algorithms (i.e. sorting objects where the only information you have about them is whether one object is "bigger"/"smaller" than the other. For objects where you have more information about the object (i.e. integers) you can sort faster. My supervisor for my Master's Thesis did just that: http://www.drdobbs.com/184404062

yes its an engineering exercise.