How will the logo and design for the site be chosen, should it make it out of the public beta? This is the first Stack Exchange site I've actively participated in, please excuse my ignorance.
That's still months into the future, and only if the site takes off.
What happens is that one day Jin will post a question titled “Design Ideas for Science Fiction site”, like the latest launch or the next launch. Then people will comment, and after maybe one or two more proposals the site will launch.
(Of course people are free to toss ideas around on Meta before that! It's #5 in the 7 essential questions of every beta, though it's a lot less urgent than deciding what's on-topic or coming up with good tags, and the promotion questions.)