One word about annualizing: what does it tell you about the coming year: nothing. Markets will change in 6 months and even more in 12 and any volatility today annualized does not tell you anything about the year to come.
This is even more true for expected returns. If markets shot up 10% last month then I bet the market will not be up annualized 120% by the end of the year.
The only thing that annualizing is good for is making numbers comparable. Calculating volatility on a weekly basis and annualizing by square-root of 52 will give something similar to the monthly data annualized by square-root of 12.
There is something more to say about data frequency (why annualied monthly vol will be smaller) but I can not go into this now. All in all annualizing makes frequencies comparable.
Another thought about SR and horizons:
If you look at short horizons, then noise will dominate the signal. You often have a bad risk/reward ratio short term. As time goes on a positive trend in your investment can show up and risk-return gets better.
You can do Monte Carlo simulations and illustrate the above phenomenon. My conclusion: improved Sharpe ratio as the horizon increases can be a good model for reality.
edit: to address the issue that if you change the risk measure to variance the picture changes: vol is in terms of the returns (square-root of sum of squared returns), variance is in terms of returns squared. This does not go together well in risk/return. What one could do is look at mean return/expected shortfall or mean return/drawdown. This is what people actually do.
EDIT 2: You say that using variance as risk measure changes the picture. Risk measures are categorized as having certain properties. One is positive homogeneity. This means that if $\rho$ is the risk measure of a random return $R$ and $h>0$ then
\rho(h X) = h \rho(X).
We can see that volatility is positive homogeneous while variance is not. For more properties see here.
In the literature about risk measures the consequences of lacking certain properties are derived. Using a risk measure that is not homogeneous (as variance) can have disadvantages.