Is what happened between the questioner and her neighbor's friend—writing an 'urfi marriage contract as a joke—considered a valid marriage, and does it entail sin or expiation after tearing up the paper?

What transpired between you and the woman is not considered a marriage, due to the absence of a guardian (wali) for the woman. Marriage without a guardian is invalid. You are not liable for any penalty or expiation if nothing occurred between you two except talk or signing, etc. However, you must seek forgiveness and repent for sitting with non-mahram (definition) women, as this leads to falling into the forbidden, bearing in mind that a jest about marriage is taken seriously.