This partial answer just addresses the UFD question. My favorite ideal-theoretic characterization of UFDs is that a domain $R$ is a UFD if and only if every $t$-closed ideal of $R$ is principal.  The *$t$-closure operation* on the fractional ideals of a domain $D$ is given by $$t: I \longmapsto I^t = \bigcup\{(J^{-1})^{-1}: J \subseteq I \mbox{ is a finitely generated ideal of } R\},$$
where $J^{-1} = (D :_{Q(R)} J)$ and $Q(R)$ is the quotient field of $R$.  The $t$-closure operation is a useful closure operation on the fractional ideals of a domain $R$.  Krull introduced such *$'$-Operations* in the 1930s in his book *Idealtheorie* and some subsequent papers.  Today we call them *star operations*.

A UFD is equivalently a Krull domain with trivial divisor class group. A Krull domain is equivalently an integral domain $R$ such that $(II^{-1})^t = R$ for every nonzero ideal $I$ of $R$.  It follows from these two well known results of multiplicative ideal theory that a UFD is equivalently a domain in which every ideal $I$ such that $I^t = I$ is principal.  I like this characterization because it doesn't mention anything about prime ideals.  After all, it is easy to see that a domain $R$ is a UFD if and only if every nonzero principal ideal is a product of principal prime ideals.  If you're allowed to mention principal prime ideals in the characterization, then such characterizations are easy to come by.

I would note that the notion of a principal ideal doesn't have a "purely" ideal-theoretic description, since they can't be recovered from the ideal lattice alone.  In particular, in my view even the notion of a PID does not have a purely ideal-theoretic description.  However, the notion of a Dedekind domain does: a domain $R$ is a Dedekind domain if and only if $II^{-1} = R$ for every nonzero ideal $I$ of $R$.  This all hinges on what you mean by ``ideal-theoretic.'' To me it means, can the notion be defined in terms of the ordered monoid of all ideals, or fractional ideals, of the ring.