After you've purchased the house, every dollar should be redirected somewhere other than the mortgage in this scenario. It's a pretty trivial difference either way, though, in terms of the actual realized gain when compared to investing that cash somewhere else. It's a second or third order component.
That's true in some scenarios, but it depends what the investment is. If someone else is paying the mortgage in a rental, for example, then it makes sense to keep the mortgage as long as possible.
With 20k invested in a 100k house that rents for 1k/month, the realized earnings are based on the initial 20k regardless of what it costs to maintain the property and loan. In that scenario, the monthly payment is $380 (30y @ 4$), taxes and insurance would be roughly $225, and then you're done. The remaining $395/mo is earned and depreciation shelters quite a bit of it. Over the course of a year, you made $4.7k from rent and another $1.4k from principal reduction. That's 30% return in the first year on 20k minus taxes, which will probably reduce it to 25%. I'm renting three houses at ratios similar to this, but I also paid more money up front for repairs. The calculation is ideal, but even cutting it by 70% - an arbitrarily huge number - still puts it way, way above the cost of the mortgage. Once you add asset appreciation (which is still based on the original 20k) and increased rent (roughly 1.8% per year), it gets better and better.
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