Let me put together the previous two answers (plus epsilon) to give an answer to all three questions.
Step 1: By Gilmer's theorem, a field K has all its subrings Noetherian iff:
(i) It is a finite extension of mathbbQ, or
(ii) It is an algebraic extension of mathbbFp or a finite extension of mathbbFp(t).
Step 2: Suppose K is a number field which is not mathbbQ. We may write K=mathbbQ[alpha] for some algebraic integer alpha. Then R=mathbbZ[2alpha] is a non-integrally closed subring of K so is not a Dedekind domain. So the only field of characteristic 0 which has every subring a Dedekind domain is mathbbQ, in which case (by the previous question) every subring is a PID.
Step 3: Suppose K has characteristic p>0. If K is algebraic over mathbbFp, then every subring is a field, hence also Dedekind and a PID. If K is a finite extension of mathbbFp(t) then it admits a subring of the form mathbbFp[t2,t3], which is not integrally closed.
So the fields for which every subring is a Dedekind ring are mathbbQ and the algebraic extensions of mathbbFp. For all such fields, every subring is in fact a PID.
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