For your second point – there are legitimate uses for other summary statistics (mode, median), but they can still be very misleading. For example, you mention using the mode of the distribution as a predictor of what the next customer will pay – this definitely wouldn't work for a company with a metered pricing plan, for example. Distributions are often not well characterized by singular summary statistics, Anscombe's Quartet (mentioned elsewhere in this comments section) is a good example of this: http://en.wikipedia.org/wiki/Anscombe's_quartet
Re: appropriate averages. Your case (metered billing) is an interesting one. I don't see why the mode is necessarily wrong - the next customer is most likely to spend the amount that is currently your most popular amount). However, in order to calculate a mode, you likely would want to bin your amounts into ranges so that $10.11 isn't treated as distinct from $10.12, etc.
You're definitely right about one summary statistic not being sufficient. I would advocate for visualization and summary statistics, with some estimate of your confidence in the estimate displayed visually.
First, I don't see how it's true that data with a relatively large amount of variance will tend to be power law distributed. Defining what a "large amount" of variance is is tough (it depends on your intuition and choice of variance metric) but there are lots of distributions with considerable variance that are, for example, normally distributed (many more than are power law distributed, as far as I can tell).
Second, if you find that this is misleading your projections, why not just use a different kind of average? For example, if you just want to know, "How much is the next customer likely to spend?", you might use the mode. Or, if you want a more robust average (i.e., less likely to be seriously thrown off by outliers), why not use the median? You can even complement these with confidence intervals if you want to get a sense of their precision.
Like twic already said, you need some indicator to understand what's going on with your business. I think that in many cases, this will be the mean. But if you want something more robust or more practical, perhaps the median or mode might suit you better.