Is there a similar address for moderators of this site? If not, would creating such e-mail (and putting contact information at some visible place) be useful?
There is one significant difference between MathOverflow and us which makes it much, much more useful for them to have a moderators e-mail account. And that is the fact that the MathOverflow brand is not owned by StackExchange and that there is a kill switch in the agreement between MO and SE for hosting that allows MO to pack-up everything and leave if they so choose. The directors for the non-profit that runs MathOverflow include the original moderators. These all add up to it being very useful for them to have a channel of discussion, which is permanently archived, which is accessible by e-mail, and which is not through the StackExchange platform.
In contrast, for Math.SE there is less call for having such a platform (in fact, I cannot think of a reason why we need to have a communications platform outside of the SE system). Currently, for discussion between moderators, we heavily rely on the Moderators only chatroom. This automatically archives all discussions, is somewhat searchable, and has the advantage that any and all newly elected moderators will immediately gain access to past discussions. (Compare this with an e-mail platform where the archive of discussion would largely reside within our own inboxes.) And for individual contact with users we have now come to almost exclusively rely upon the "Contact User" form in the SE network. Again, this method has the advantage of automatically archiving all communications and additionally increasing transparency (as all moderator messages will result in notifications being sent to all moderators).
From personal experience having served as a moderator during the era before chat.SE came into being and when all moderator decisions are coordinated via e-mail, I have to say that
- The current set of tools given to us by SE has a small learning curve and does take a little getting used to...
- ... but after the initial period I definitely much prefer the current system.
That said, if you really must contact the moderators in private outside of the SE network: several of the moderators (yours truly included) have an e-mail address listed in the About Me section of their profiles. But note that we may move the discussion to an SE-archived venue if the situation calls for it.
No, there is no such mail address for this site. Using email for moderator purposes has certain drawbacks, and SE tries to keep all moderator communication inside the SE platform for that reason. Keeping this all inside the SE software makes it much easier to see what happened afterwards, which is very useful if there are any complaints about the issue.
In most cases, you don't need to contact all moderators, you need to contact any moderator. And that is what flags are meant for. For the rare case where the issue is too complicated for a flag, you can use the "contact us" link at the bottom. This goes to SE, but they can forward the information to the moderators or handle it themselves. Anything that doesn't have to be private should be on meta anyway.
Suspensions are always accompanied by a moderator message (this is enforced by the software), and the suspended user can reply to the moderator message. They can only reply once to each moderator message though to prevent abuse or spamming of the moderators.