What happens if a transporter transes The Godfather so that he targets the mafioso, does the mafioso kill himself (since the GF orders the mafioso to kill his target), or does the gf kill the mafioso?
The Transporter does not change the targets of the people they choose each night. You may be thinking of the Witch.
The Witch can absolutely control a Godfather into a mafia member (or the Mafioso into the Godfather), though it's against their interests as the Witch wins with the Mafia.
The Transporter can swap the Godfather's target with a mafia member, causing the mafioso to possibly kill himself if the transporter gets lucky.
For example, if Godfather targets the Medium, but the Transporter swaps the Medium with the Mafioso, the Godfather will order the Mafioso to attack the house now occupied by the Mafioso; the Mafioso will kill himself.
When a Witch or Transporter causes the Godfather to get attacked, his Basic defense does not register - all attacks from his own mafia will kill him.