In "my site", under the "Content" tab, there is the "SharePoint documents" webpart.
How does this webpart works ?
I'm wondering if I can reuse this webpart to show global task lists or, at least, get from the profil the sites where I belong.
The content area of a user's MySite is in fact a personal website for that user (technically it is actually a site collection).
Every user has their own Content website, and inside each of those sites by default is a document library.
So what you are looking at is just your own content website, and your own document library. If you add a document here it lives locally in your website, not in some global document repository.
Every user is an admin of their website so they can add any webparts that you have given them permission to. For example if you create a 'global tasks' web part and make it available in the web part gallery, then any user can add that web part to their own content site.
If you want to create a customised web part such as 'global tasks' and have it available on everyone's content site when they log in automatically, then this is much more difficult, but achievable - you need to use feature stapling most likely.
We were also wondering how the 'Sharepoint Documents' web part under each employees' My Content tab of their profile worked. Mainly because we found it wasn't always showing everything properly. For example, under the 'Shared Documents' it wasn't showing everything in the Shared Documents for that user.
We figured out that this web part will ONLY show documents that were 'Last Modified' by the user. We didn't dig into how it actually functions but that is the basic jist of what shows there.
Hope that helps