We often don't know the underlying distribution of a particular process or our data, true, but that doesn't mean that we can't get useful information out of it by making some assumptions, provided that our assumptions are not wildly wrong. For example, if we assume height is normally distributed, that should mean it's possible, under our model, to see someone who is -9000 feet tall. We're willing to accept this falsity in exchange for having some nice properties we wouldn't have had otherwise, provided we exercise some common sense.
In regards to (3), if all you can wield is the SLLN, you can't do much inference or prediction. So you're able to guarantee that the sample mean will converge to the expectation almost surely. The closeness of approximation in a finite sample case using asymptotics is intrinsically tied to sample size and the particulars of the underlying distribution of the random variable.
In regards to (5), the curse of dimensionality does become a large problem, but throwing out assumptions makes it a far bigger problem. The nonparametrics community has a lot of trouble with multivariate distributions for precisely this reason! This is one area where parametric models do better than non-parametric ones, since they have so much more structure to play with, it makes the problems so much more tractable.
You mention order statistics... order statistics in higher dimensions is a tricky concept that is not as defined as the one dimensional case. Which one is more: (3, 2, 1) or (1, 2, 3)? Wouldn't that necessarily depend upon the underlying distribution or some other measure of distance? If you have a paper to suggest that general high-dimensional order statistics makes sense, throw it this way. You are correct that Least Squares doesn't require a distribution to get mean and variance estimates, but what good is that if you don't know how much of your population fits within a multiple of your standard deviation around your mean? You could use something that works for all distributions like Chebyshev's Inequality, but to get that level of sweeping generality seriously hurts your power (If your data IS actually normal, 95% of obs fall within 1.96 SD's of the mean. Chebyshev will tell you 75%), and even Chebyshev had to impose a restriction to get that bound--it assumes finite mean and variance.
While there are distribution-free hypothesis testing methods, most of classical nonparametric statistics makes some assumptions about the underlying distributions, albeit they are far gentler than the parametric assumptions.
Always choosing nonparametric methods is just as short-sighted as always using parametric methods. It's very possible that you are throwing away vast amounts of information by using a nonparametric procedure when a parametric one would have been completely adequate. With ANOVA and two-sample tests, you don't lose much by going with Kruskall-Wallis or Mann-Whitney tests (under normality, MW has a relative efficiency of 3/pi versus the t-test), but in other circumstances, using a nonparametric method could be way worse, provided the parametric assumptions are true.
What I'm getting at is that while non-parametrics is nice in that it doesn't require many assumptions, they may be throwing away too much. Fitting to a distribution may very useful in making inferences and predictions. All models are wrong, but some are useful.