Maybe this plugin goes in the direction you are looking for: http://community.elgg.org/plugins/797106/2.1.1/digest. But it sends notifications / email to all users and not admins only. Also, I don't know what exactly is included in the notifications. But the plugin might be a starting point to customize it regarding your requirements. At least a daily summary would surely be better than a separate notification about every single activity which would flood your inbox. But isn't the site activity page also a way to keep up with the site activity without getting emails abut every single action?
Not in the default system log.
I haven't found the default system log useful for *anything*. I routinely disable it on all projects as it just adds overhead. I would suggest you create a custom log of only what you need.
Since i am new to developing for Elgg, i am a little lost on how to create my own system log. Has anyone already perhaps developed a configurable plugin for logs.
I don't think there is a plugin that offers a configurable system log.
Take a look at the file engine/lib/system_log.php. The functions in it handle the creation of the system log entries. What you would need to do within your own plugin:
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