Yes.  Whether you'll win depends in large part of the terms of the warranty.  I suggest you review the full warranty document carefully before doing anything.

I'm not really sure what this question means.  You can say this at any time.

I agree.  If it were my car, I wouldn't have let the situation persist more than a couple weeks.

I don't really see the point in sending the letter by certified mail.

Keep in mind that the dealer likely isn't the obligor under the warranty.