Is a university that grants me a PhD for $1000 and a copy of my unpublished book fake? I have paid $1000 for PhD in mathematics. And I have sent them my book to be considered as a PhD thesis.

Now I suspect that I have lost the money.

But as far as I know they are not accredited by the U.S. Department of Education.

So, is it possible that I will find a job requiring a degree, using this (fake?) PhD?

Now they have told me that the U.S. Department of Education will send my diploma to US foreign affairs department and then they will make some "stumps" on my diploma. Do these stamps mean anything? Are these stamps just a formal thing which mean nothing and don't make my degree "official" ("real")? Moreover, to send my diploma to the departments costs additional $1000-$1750 (dependent on whether I pay right now or later).
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 Worse still, even if you were to get an offer, it could be rescinded when the truth is uncovered.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.