If we propose a topic on area51, and it enters beta, can we use a custom domain and logo for it? Or are we only able to use the topic-title.stackexchange.com domain? Also, do we get the ability to edit the topic we proposed?
Nope, not usually. It only happens if you have a really popular site, with support from the "man" (Stack Exchange Employees). However if you were referring to using your own domain, nope, it will never be your site.
However if a site gets out of beta mode, the community will choose logos, and such.
For more on why domain names are rarely used, please view this.