In Chapter 19, Cole apologizes for his attitude... how does Edwin respond to his apology, does he believe it is genuine?

No, Edwin does not believe that Cole's apology is genuine... he believes that Cole's apology is nothing but noise, a tactic to get his own way.

“I said I was sorry,”Cole said.
“You also said you had changed,”answered Edwin.
“Please don’t take me back.” Cole’s voice wavered.
“I promise I’lltry harder. I’ll do anything you ask.”
Edwin stood and faced Cole across the fire.“It’s time to cut losses and send you home.”
“I didn’t mean what I said. I just—”
Edwin held up his hand. “Stop your mindless talking! Your words insult me! They’re just noise in the air— they don’t mean anything."