After xyz signups through it, make the rest of the day's accounts require manual approval for the first few posts.
You can also mitigate the disposable mail problem pretty easily by requiring whitelisting for any domain that's got more than a few accounts tied to it.
That comes out way cheaper for you, and might improve the quality of conversation (because there are valid reasons for anonymity and pseudonymity).
EDIT:
Also, bug report: the category row at the top doesn't overflow very well; on smaller window-sizes, there are many categories out of view. Maybe make the hamburger menu come up on more than just mobile screen factors?