Is what happened between the questioner and her friend—writing an 'urfi marriage contract as a joke and then tearing it up—considered a legitimate marriage that necessitates expiation or incurs a sin?

It is not permissible to sit and converse with foreign (non-mahram (definition)) women without necessity, due to what it may lead to of falling into the forbidden, such as looking and deriving pleasure from voices. This is to block the means to falling into the forbidden. What transpired between the two of you is not considered a valid marriage, as the validity of marriage requires the presence of a guardian (wali) for the woman. Every marriage without the permission of the woman's guardian is null and void, based on the saying of the Prophet (peace be upon him): "Any woman who marries without the permission of her guardian, her marriage is null and void, her marriage is null and void, her marriage is null and void..." and his saying: "There is no marriage except with a guardian," and his saying: "There is no marriage except with a guardian and two just witnesses." If nothing transpired between you except conversation and signing, then no compensation or expiation is required. However, you must seek forgiveness and not return to such gatherings, knowing that the jocularity of marriage is considered serious, based on the saying of the Prophet (peace be upon him): "There are three matters whose seriousness is serious and whose jocularity is serious: divorce, marriage, and revocation (of divorce)."