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Isn't there a mod or add-on that notifies the board admin when any member creates a topic or post?

Maybe through email? If there is, please provide a link. If there isn't, this is great suggestion

For new topics you could try:
http://www.elkarte.net/community/index.php?action=profile;area=notification
select all boards and "update".
For all replies... did we ever manage to do something like that? I remember some discussion, but I don't remember the outcome...

ohh, sorry, is this provided for all members or only the administrator? Thanks!

You can control who has access to this feature with the user group General Permissions ... the settings should be under the "Notification" area of the general permissions. 

You can get topic or all reply notifications ... full text of the post or just a simple message, your choice.

lol, thanks man!
Pls how do I pay to remove fotter credit

Could a similar thing be done with notifying on a new forum post to an IRC channel? Would be a nice add-on to include as part of a community, since I run an IRC channel for my community and it'd be nice to keep things updated for the users, since they mostly hang around there.

hmm... it would require an IRC bot somewhere in the process at least...

Like SMF, ElkArte got rss if that can be useful to you. During twitterfeed.com time (now no more), forum owners used to post to social media using it.

For IRC you can check this forum link if its given script using rss in there does what you need. I don't really comprehend it and not really sure whether its relevant too but it came out from googling around for answers.