Sometimes I get email notifications that a friend has shared a document with me. But I don't want to be part owner.
How can I refuse to be the document's owner? Can I refuse when someone is trying to share a document with me?
You're the document's owner when you create that particular document, if you want to stop being the owner you might want to delete it. On the other hand, if you want to refuse the sharing of a document created by someone else you might as well delete it from your list (by clicking on the checkbox at the left-side of the document) and then click Remove at the top. Hope this clarifies your question.
If a person is maliciously adding you as an editor (or owner) to documents, you could report the Google Drive/Docs abuse to Google.
If it's a friend who's behaving anti-socially, tell him or her why you don't want to share/own the document. Unfortunately, in the email that's generated, I can't see any way to "refuse" the share in the email. Google Docs is not a social network, per se, where you'd find such features (ignoring requests).
From the tests I did briefly, I can only assign one "owner" of a document. An editor of a document can remove himself from the sharing. If you're just an editor, deleting the document won't remove you from having the permission (although the document doesn't show up either in the Trash or the All Items).