Is a university that grants me a PhD for $1000 and a copy of my unpublished book fake?
You have lost the money whether or not the "university" gives you the degree.

You have given your money to what is known as a diploma mill. A long time ago (nearly two decades!), I received an email offering degrees from "prestigious, non-accredited universities" based on life experience. This is a complete and total scam. Even if you have a diploma, it will be useless for professional purposes. Anybody responsible for hiring a PhD will see that there is no work resulting from your "graduate career," which will be a tipoff that the degree is worthless, and you will be unlikely to receive a job offer. Worse still, even if you were to get an offer, it could be rescinded when the truth is uncovered.

Disengage now, before you lose any further money on this situation.

Addendum: I should also mention that I am unaware of the US Department of Education doing any certification of diplomas and certificates on an individual basis.