Trying to understand Search Box configuration

After an hour of bouncing around the forum and trying various solutions to get the Search Box to both appear and appear where I wanted it, I have to say that while I've gotten it where I want it, I have NO idea how all this works.

Page in question: http://www.todddawsondesign.com/test/matia/
ATA: 3.7.9
WP: 3.4.2

It seemed like when I had the Search Box without Absolute Positioning, it didn't show up-it seemed the header image/logo was blocking it. Are they in the same div? On the same level? Does one block the other? How would I get the box to show on top of the logo? I did delete the RSS and social feeds (a choice of this showing or not would be a nice addition to future iterations).