1) To be honest, any horizon is problematic in this respect. Simple sampling statistics 101 will tell you that the standard error around any estimate of true mean returns is the root time * variance. So for eg stocks at 20 vol, that's a +/-40% 1y 95% confidence interval around your sample mean ;-) With 100 years of data, that's still +/-4%! Which is in-line all too many estimates of the equity risk premium...
The problem here is as much as methodology as your sample, because the classic problem with the mean-variance optimisation approach from the get-go is that is hugely sensitive to the input assumptions. A couple of percent different on the returns and you get a very different portfolio output suggested.
Volatility and correlation regimes also shift over time; but in a sense, that makes it more OK to use shorter-term assumptions for these than for the returns. Because it's more likely their recent behaviour reflects the current paradigm; and these often do stick around for a while.
2) It's easy to calculate the GMV portfolio. It's simply the Max-Sharpe Portfolio if you assume equal returns across all your assets. Then MinVar becomes MaxSharpe! Likewise, assume returns proportional to volatility (ie equal Sharpe across your assets), and Max Diversification becomes MaxSharpe. Seen from the other side, Max Sharpe using last 12m returns is nothing more than a "long Momentum" portfolio.
