I see in the preference page of the error reporting, that I can change the address of the server. Is it possible to host the server locally?
You can point the client to any URL using the preferences. You can also set this value via eclipse.ini or system properties. For SR1 we'll likely make this a little more convenient for external parties.
Where can I find the source code of the aeri server?
There is a perl script in EPP logging repository which takes requests and dumps them to disk. You can use it as baseline for your own implementation. The service at eclipse.org is strongly interwoven with the eclipse.org infrastructure and not reusable outside of Eclipse. As such it's not open-source. If you are looking for services, you may contact Codetrails directly.
is it possible to track a non eclipse plugin errors using the AERI?
If you looked into the preferences/code you will notice that by default it only logs errors logged by Eclipse.org plugins. This can be customized by modifying the preferences (e.g. in eclipse.ini). Check the PreferenceInitializer class for details.