Wow! Hard to argue with any of that.Your wife should be calling you instead of her Dad. She’s undermining your role and damaging your relationship. She owes you an apology.
Your FIL did his best and the cheap jack failed. You owe him an apology.
You let your wife drive a car with a bad spare. That’s your failure. You owe her an apology.
You lost your temper in front of them. You owe them both an apology.
Man up, own your mistakes, all of them, start apologizing.
Then, talk with your wife about who she called and why.