So it sounds like a lot of people are ready to make a go at this. Setting up the servers / VMs to host it is really out of my league, but there are a few points about the wiki that might be useful to keep in mind.
The main one (and a bit of an obstacle) is that I don't think there's a simple way to get backups through the wiki itself. There's a "package" action available on a page-by-page basis, but I think it only takes one or a few page-names at a time (that you have to manually enter). I can't vouch for how much drain it puts on the server and what it spits out either. Honestly, if you can't get in touch with Dimi about a tarball of everything, your best bet may be to sick a crawler on the wiki, then run the actual pages through something like Beautiful Soup.
If you do manage to get an archive of the entire wiki, logs and all, you might find the scripts I worked on a few years back helpful, more as a testing & knowledge record of how MoinMoin actually stores everything. I'd say they were mostly finished (except maybe bcrypting user files) and ~85% tested; so if you do have a direct archive of the actual wiki and want to try running them, just be sure to keep a backup copy of all the wiki data in case they act up: https://bitbucket.org/kauble/moin-admin
One last but less immediate point is that while I don't know the details, I think MediaWiki has actually become much more flexible about theming over the past several years. The Python components of the current wiki theme are pretty MoinMoin-specific, but I tried to make the CSS files as simple and similar to the rest of WineHQ as possible. You can probably reuse those and any images or icons for MediaWiki without too much work: https://bitbucket.org/kauble/wine-wiki-migration
Kyle