Stylized Facts: Price Dispersion
A little empirical evidence for your first point can be found in


Erik Brynjolfsson &amp; Michael D. Smith (2000): "Frictionless Commerce?
A Comparison of Internet and Conventional Retailers", Management
Science, 46(4), 563–585.


They look at the markets for books and for CDs both online and offline and estimate the price distributions. Whilst they look at price changes, they do not present data on the change in price distributions over time.