We study Néron models of abelian varieties over the arithmetic base $S=\Spec\mathbb{Z}$. (1) \emph{Isogeny extension \& component control:} prime-to-residue-characteristic isogenies extend over a dense open, act finite étale on identity components, and induce component-group maps with kernel/cokernel annihilated by the isogeny degree; local Tamagawa ratios are uniformly bounded. (2) \emph{Semicontinuity:} for $p\nmid m$, the function $t\mapsto v_p(\#\Phi_{A,t})$ is upper semicontinuous on a common open locus, and component indices across degree-$m$ prime-to-$p$ isogenies are uniformly $m$-bounded. (3) \emph{Cohomology \& conductors:} on the tame locus, $I_p$–invariants in $H^1_{\mathrm{\acute et}}$ are isogeny-invariant; conductor jumps are quantitatively controlled by component-group indices. (4) \emph{Hecke-uniformity:} For integral models of modular curves $X_0(N)$ over $\Z[1/N]$, the prime-to-$p$ Hecke correspondence $T_\ell$ ($\ell\nmid Np$) extends as a finite étale correspondence inducing $\ell$--power isogenies on the associated Néron models over a dense open locus. These isogenies act finitely étale on the identity components and yield morphisms of component groups whose kernel and cokernel are annihilated by a bounded power of~$\ell$. Consequently, the toric ranks, inertia-invariant cohomology dimensions, and local conductors remain uniformly controlled and vary within an $\ell$--power-bounded range along each $T_\ell$–orbit. In the Hodge-type Shimura setting, an analogous uniformity holds after removing finitely many primes of bad reduction: every prime-to-$p$ Hecke correspondence decomposes on a dense open into prime-to-residue-characteristic isogenies of abelian schemes, finite étale on the identity components, with component-group variation uniformly bounded by the degree of the correspondence. On this Hecke-uniform locus, the induced maps preserve inertia-invariant cohomology and bound the variation of local conductors, giving orbitwise constancy of toric rank and controlled arithmetic variation compatible with the geometric and cohomological invariants of the Néron models. (5) \emph{Heights:} On the semistable locus, for abelian schemes equipped with a fixed symmetric ample line bundle, the line of invariant differentials $\omega_{A/S}$ provides a canonical measure of positivity. This induces lower bounds for Néron--Tate heights that remain stable under all prime-to-$\Sigma$ isogenies within the family, expressing that the arithmetic height and the geometric degree of $\omega_{A/S}$ vary in a uniformly controlled way.