When I saw the Badge design (next to the Badge Count in the header) on this site, I noticed it was normal. I would think it would be at the very least interesting to have a LEGO brick there instead. I know it is possible, other SE sites have Badge Designs related to the site (Christianity.SE for example has a Christian Fish as the Badge design in the header). What can be done to change this? Who can do it? May it be done?
It can indeed be done - however currently we're still officially in beta (hence the "Beta" after our site name in the header), and so don't qualify for a custom skin or the other changes that come with a fully graduated site.
In terms of actions, one of the things that Stack Overflow is keen to do is reduce the impact of design-by-committee - which has historically been an issue, and they do have in-house designers that take care of this, but there are a lot of sites now.
In fact, if you sort that list by age, you'll see we're the first graduated site with our own logo instead of just our initials in a blue speech bubble.
We'd probably also want to be distinct from the Gaming badges which look like 1x1 round studs.
In terms of what we really need to do, we need to consistently get at least 10 good quality questions day, which would push us closer to graduation and then consideration for a custom site skin.