The above comment with $n=2$ readily adapts to the global number field setting, using orders with $p$-power index in a quadratic extension of $\mathbf{Q}$ unramified at $p$.

To make a local counterexample, let $A$ be an unramified extension of $\mathbf{Z}_p$ with degree $n > 1$ and $R \subset A$ the subring of elements reducing into $\mathbf{F}_p$ in the residue field $A/pA$. Then $R$ has maximal ideal $pA$ that is not $pR$, and $pA/p^2A$ as an $R$-module is "just" an $\mathbf{F}_p$-vector space of dimension $n$. Hence, one can pick $I, J, K \subset pA \subset R$ corresponding to subspaces $V,V',W$ of $\mathbf{F}_p^n$ such that $(V+V')\cap W$ strictly contains $(V\cap W) + (V'\cap W)$. Three distinct lines through 0 in a plane does the job when $n=2$.

More generally, real-analytic geometry is much closer to $C^{\infty}$ geometry than to complex-analytic geometry due to Grauert's theorem that every real-analytic manifold is a real-analytic submanifold of a Stein space (and the Morrey-Grauert theorem that gives a unique real-analytic structure to any smooth manifold). The question above is one of many consequences of Grauert's results (which should be more widely known); see the last Theorem in the Introduction of this 1981 paper: archive.numdam.org/ARCHIVE/CM/CM_1981__43_2/CM_1981__43_2_239_0/…

Or see EGA $0_{\rm{IV}}$ 19.5.4 with some extra topological pizzazz (that can be ignored, using discrete topologies). They use the older technique of extensive cross-referencing to substitute for the lack of search functions in physical books. :)

OK, this is what I think is usually called the "rigidified" Poincar\'e bundle (typo: you missed an inversion on the 2nd tensor factor). But encoding such rigdification is part of the very content of building a Poincar\'e sheaf on the entire Picard scheme (or algebraic space), so I remain puzzled as to where the point of confusion is arising for Pic$^0$ versus Pic in terms of Poincar\'e bundles (i.e., if you are happy for Pic then why not for Pic$^0$?).

The description of the bottom row is "cheating" a bit, since in reality one should say we use the isomorphism $\mathscr{Ext}^1(A,\mu_{\ell^n}) \simeq \mathscr{Ext}^1(A,\mathbf{G}_m)[\ell^n] = A^{\vee}[\ell^n]$ and the pairing between torsion-levels in $A$ and $A^{\vee}$. (Edit: answer_bot wrote a related comment at the same time.)

If you have a given abelian scheme $B$ with line bundle $L$ on $A \times B$ equipped with trivialization $i$ of its pullback to $A \times \{0\}$ then to check if the resulting map $B \rightarrow A^{\vee}$ is an isomorphism (thereby giving the universal property to $(B, L, i)$ it suffices to check on geometric fibers, where various results in Mumford's book are applicable. I don't know what "modified Poincar\'e bundle" means (FGA Explained not nearby at the moment), but would that address whatever is concerning you?

The Poincar\'e bundle is tautologically part of the very meaning of "dual abelian scheme" or representability of the (rigidified) Picard functor (say as an algebraic space, which in turn is a special case of Artin's theorem on Picard functors).

Oort's book "Commutative group schemes" has a very nice discussion of both the representability of functor $T \mapsto {\rm{Ext}}^1_T(A_T, {\mathbf{G}}_m)$ by the dual abelian scheme when the latter exists (which is always the case, by the result of Raynaud) and not only the relation of its $n$-torsion with Cartier dual of that of $A$ but also the more subtle issue of relating double-duality on both sides. Oda's paper addresses the double-duality aspect (and much more) in its first section.

The functor ${\rm{Pic}}^0_{A/S}$ is a subfunctor of ${\rm{Pic}}_{A/S}$ (defined by a condition on geometric fibers), so what is the meaning of the question in the "Edit" that isn't a tautology (via pullback)?

@thierrystulemeijer: On character groups the map is an inclusion $\overline{L} \rightarrow L$ between finite free $\mathbf{Z}$-modules of rank $p-1$ such that the cokernel is cyclic of order $p$, so in suitable bases it looks like the direct product of the identity map on $\mathbf{Z}^{p-2}$ and the index-$p$ inclusion of $\mathbf{Z}$ into itself via multiplication by $\mathbf{Z}$. Now apply ${\rm{Hom}}(\cdot, {\rm{GL}}_1)$ to turn that back into a map of tori. So all I am doing is "choosing good coordinates" for describing the map on tori. I don't understand your proposed description, sorry.

@KeerthiMadapusiPera: Is $I$ defined as the identity component of the intersection? I think one has to control the entire intersection, and its centrality in the entirety of $G$ has to be proved. The "problem" is that the good behavior of centrality under quotients, a familiar feature of connected reductive groups, is not true for connected linear algebraic groups more generally, so at this early stage in the theory some care is needed for handling centrality claims.

@KeerthiMadapusiPera: I think your suggestion is circular, since the classification with Dynkin diagrams (which, oddly enough, is not discussed in Borel's textbook) requires a huge amount of structure theory, the entire foundation of which rests on knowing the classification in semisimple-rank 1 to get a root system...

This is proved in Mumford's book on abelian varieties in section 22 (see "First Example", and the Theorem there). As in Mumford's discussion, the notation should keep track of the choice of CM type $\Phi$ giving the identification of $K \otimes_{\mathbf{Q}} \mathbf{R}$ with $\mathbf{C}^d$ as $\mathbf{R}$-algebras; this is implicit to make sense of the quotient of $\mathbf{C}^d$ modulo orders in $\mathscr{O}_K$ or fractions ideals of $K$ (though since the CM order is not preserved by isogeny, it might be "better" to avoid formulating things in terms of orders).