The difference between your preferences over choosing lottery A vs. lottery B when both are performed a million times, and your preferences over choosing A vs. B when both are performed once, is a measurement of your risk aversion; this is what Gray Area was talking about, is it not?

No, it's not, and the problem asserted by Allais paradox is that the utility function is inconsistent, no matter what the risk preference.

Then you must be using a different (and, I might add, quite unusual) definition of the word "preference". To quote dictionary.com:

1. to set or hold before or above other persons or things in estimation; like better; choose rather than: to prefer beef to chicken.

I don't see anything in there that about how many times the choice has to happen, which is the very issue at stake.

If there's any unusualness, it's definitely on your side. When you buy a chocolate bar for a dollar, that "preference of a chocolate bar to a dollar" does not somehow mean that you are willing to trade every dollar you have for a chocolate bar, nor have you legally obligated yourself to redeem chocolate bars for dollars on demand (as a money pump would require), nor does anyone expect that you will trade the rest of your dollars this way.

It's called diminishing marginal utility. In fact, it's called marginal analysis in general.

What does it mean to say that you prefer B to A, if you wouldn't trade B for A if the trade is offered?

It means you would trade B for A on the next opportunity to do so, not that you would indefinitely do it forever, as the money pump requires.