As for the chasing effect, if you put the pins in groups of two, or three, it would still look like a chaser, but only need half (or a third) of the pins.
So you are saying just have 2 (or 3) LEDs go on at once, then the next 2 (or 3) would go on as the previous 2 (or 3) shut off and so on to create the chasing effect? Or are you saying there is some way to make each individual LED in a group of 2 turn on even though the group of two is connected to one pin? Did that make any sense? 
Nick, thanks so much for testing that out and explaining it in such depth to me. It's really making all this a lot easier to understand.
As for the power supply, would something like this work? http://www.radioshack.com/product/index.jsp?productId=10788649&filterName=Category It's 6V so the Arduino wouldn't have to burn much off, correct? Because, as I understand now, the Arduino regulates everything down to 5V, yes? 