That is TrikeApps understanding too - we pay for and maintain the servers, but we do so for Eliezer and SIAI, don't consider ourselves to have a moral right to use the practical powers we have, and generally do what Eliezer tells us to do.

So a longer form answer that completely agrees with Eliezer's claim above is…

We care about what the LW community wants, and if we thought Eliezer was being too much of a dick we could be driven to resign, but if we did that we'd carefully and responsibly hand control back to Eliezer and SIAI first. And he'd have to be a lot more of a dick than he occasionally appears to be.

Who owns the domain?

http://www.whois.net/whois/lesswrong.com, but the moral right is SIAI's (in a fairly well documented way).

Who owns the code?

Open Source, forked from the Reddit codebase. See https://github.com/tricycle/lesswrong/blob/master/LICENSE.

Who created the Wiki?
Who pays for the servers?

TrikeApps and TrikeApps, but for SIAI/Eliezer.

If someone is in charge, what rights do they reserve for themselves?

There are practical powers TrikeApps hold, but we reserve only the right to withhold our labour, and only that after handing back practical control to Eliezer/SIAI.