GPO Style Manual, among others does include it. I don’t have access to to current AP but suspect it does—what you don’t mix is honorific and suffix. Anyway, it is simply a common courtesy and respect to include Dr. or Ph.D. And it is entirely appropriate to say Dr. Biden. Complain about her politics, about her husband, whatever, but let’s be respectful of the degree that she earned. I’m sure you would have no problem doing it if it was somebody with politics you agreed with. So do it here.
Not me. I don’t recognize a pHD in education as legitimate. Besides I’m not big on massaging egos.
Lawyers earn their degree too. Do you use the honorific Esquire when you talk to or about one?