If Fred asks a friend of Bob to talk to Bob about it, and Fred agrees
(to the mutual friend) to pay 50% the initial wage, is this agreement
Probably not. The mutual friend is in a position to see that there is a basis for a settlement that both parties would agree upon, but there needs to be a communication between the parties themselves (or through the mutual friend acting as a messenger back and forth) for there to be a binding agreement.
The exact language of the communications and role of the mutual friend would matter in determining what happened from a legal perspective.
Would it make a difference if Fred is CC'd on the email where Bob
agrees to pay Fred 50%, and Fred replies accepting this as a
Yes. Then you have offer and acceptance, and the third-party is just a bystander. A slight variation of this would be common in a formal mediation.
Hypothetically if Bob decided to still not pay 50% could Fred use this
as evidence to sue?
It depends on the context which this question doesn't make clear.
Settlement negotiations are not admissible as evidence to prove liability on the merits.
But the settlement negotiations can be admissible to prove a claim that there was a settlement agreement involving an offer and an acceptance that formed a settlement agreement contract.
Mediators themselves and some mediation communications are privileged from being used as evidence in court for some purposes, but the written communications showing an offer and acceptance of a settlement agreement could probably be admitted (and, of course, a mere mutual friend doesn't count as a full fledged mediator for the purposes of this legal privilege).