Sort of a dumb question
Who does the site design after its graduation. Do we get to choose/propose designs for different widgets?
If you search through the meta of most graduated Area 51 sites you'll find a thread like this one posted by one of our designers. They'll work up a design concept and submit it to the community for feedback.
Frequently, part of the process involves the designer hanging out on meta and reading any suggestions that have been made previously, so you are welcome to float ideas. However, keep in mind that the site won't be "designed by committee" - our designers aren't obligated to incorporate ideas that come up here. However their goal is to create a design that represents this community so it's likely they'll take some of it to heart.