What is the average price per piece?
A brief search led me to a very thorough blog post by Ruth Suehle on GeekMom, which includes the data and a Graph:


  Does It Feel Like Lego Bricks Just Keep Getting More Expensive?


In her findings, she found that average cost per piece in 2011 is about US$0.12, down from a high in the '80s of about US$0.40 - these prices were adjusted for inflation, etc.

Prices tend to be most affected by the price of oil, as that's a key component in plastic manufacture, so we might expect to see prices starting to creep up again.

Interestingly the licensed sets (Star Wars, Harry Potter, etc.) don't seem to have much of an impact on the price of the sets, it's just that they are often bigger sets, so have higher price tags - For example the Super Star Destroyer with a retail price of $399 had 3152 pieces working at $0.13/piece - although yes, an extra $0.01 multiplied over 3000 bricks does seem to add an extra $30 to the set.

There doesn't appear to be much data on weight or type of brick.