What is known about ideal and divisibility lattices of GCD domains and their generalizations?
Given a ring $R$, let us denote by $L(R)$ the lattice of two-sided ideals of $R$ for which the infimum and supremum are given by $\inf(I, J) = I \cap J$ and $\sup(I, J) = I + J$.
Such lattices are complete and modular. If $R$ is a principal ideal domain (see original MSE question/answer) or if $R$ is the ring of integers of a number field [3, page 135], then $L(R)$ is distributive.
The following claims classify the lattices of principal ideal domains (PIDs).
Claim 1. Two PIDs have isomorphic ideal lattices if and only if they have equinumerous sets of prime ideals.
Claim 1 is, I believe, immediate. (In [2], it is proven that $L(\mathbb{Z}[\sqrt{-5}]) \simeq L(\mathbb{Z})$; this should give an idea as how to proceed.)
So are the following claims.
Claim 2. Let $\kappa$ be a cardinal number. Then there is a PID with exactly $\kappa$ prime ideals.
Proof of Claim 2. For $\kappa = 1$, take $R = \mathbb{Q}$. For $2 \le \kappa < \omega$, consider $\kappa - 1$ distinct prime numbers $p_1, \dots, p_{\kappa - 1}$ in $\mathbb{Z}$ and set $S \Doteq p_1\mathbb{Z} \cup \cdots \cup p_{\kappa - 1}\mathbb{Z}$. Then the localization $\mathbb{Z}_S$ of $\mathbb{Z}$ is a PID with exactly $\kappa$ maximal ideals. For any $\kappa \ge \omega$, set $R = k[X]$ for $k$ an algebraically closed field of cardinality $\kappa$. (For $\kappa = \omega$, setting $R = \mathbb{Z}$ is a simpler and natural choice.)
Claim 3. The ideal lattice $L(R)$ of a principal ideal domain $R$ is totally ordered if and only if $R$ is a field or discrete valuation ring. It is Boolean if and only if $R$ is a field.
The question as to whether $L(R)$ is complemented was settled by Robert Blair in [1, Theorems 1 and 2] in a broader setting:
Blair's Theorem. Let $R$ be a (not necessarily commutative, possibly without identity) ring. Then the following are equivalent:
$(i)$ The lattice $L(R)$ of two-sided ideals partially ordered by inclusion is complemented.
$(ii)$ The ring $R$ is a direct sum of minimal two-sided ideals.
$(iii)$ The ring $R$ is isomorphic with a direct sum of simple rings.
The question as to whether $L(R)$ is distributive for more general rings $R$ was also tackled by Robert Blair in [1, Section 5]. Here is one of Blair's results.
Blair's Lemma. (Lemma 11) If $R$ is a primitive ring whose lattice $L_r(R)$ of right ideals is distributive, then $R$ is a division ring.
[1] R. Blair, "Ideal lattices and the structure of rings", 1952.
[2] H. Subramanian, "Principal ideal in the ideal lattice", 1972.
[3] G. Birkhoff, "Lattice theory", 1948.
A PID is a unique factorization domain, so the divisibility lattice (identifying associate elements, of course) has the same ordering as the set of $\mathbb{N}$-valued functions with finite support on some (possibly infinite) set of primes, plus an additional top element corresponding to 0. Actually, I suppose it is the reverse of this in your convention, where $a \leq b$ corresponds to "$b$ divides $a$".
It is thus distributive, but not Boolean (except in the trivial case where the set of primes is empty; i.e., when the PID is a field); the only complemented elements are the top and bottom element, and indeed, if $a \wedge b = 0$ (in the sense that $lcm(a, b) = 0$), then at least one of $a$ and $b$ is itself $0$.