Is the statement: "By God, whoever sends you a message on Facebook, I will kill him" (meaning young men) considered one of the forms of repetition, given that the intention is not remembered, and does it entail the obligation of killing if the intention was "every man who sends you a message, I will kill him" and I did not kill anyone?

The phrase that necessitates repetition is "كلما" (whenever). If this phrase is not used, the oath is not repeated. The default is that the breach of an oath is not repeated with the recurrence of the action, unless the intention for repetition is stated. A single expiation for the oath is due if someone sends a message to your fiancée, and the oath is thereby dissolved. It is not permissible to expand in conversation with the fiancée, because she is a stranger (non-mahram), and it is not permissible to speak with her except to the extent of necessity and without softness in speech.