Near-additive (aka $(1+ε,β)$-) emulators and spanners are a fundamental graph-algorithmic construct, with numerous applications for computing approximate shortest paths and related problems in distributed, streaming and dynamic settings. Known constructions of near-additive emulators enable one to trade between their sparsity (i.e., number of edges) and the additive stretch $β$. Specifically, for any pair of parameters $ε>0$, $ κ=1,2,\dots$, one can have a $(1+ε,β)$-emulator with $O(n^{1+1/κ})$ edges, with $β= \left(\frac{\log κ}ε\right)^{\log κ}$. At their sparsest, these emulators employ $c\cdot n$ edges, for some constant $c\geq 2$. We tighten this bound, and show that in fact precisely $n^{1+1/κ}$ edges suffice. In particular, our emulators can be \emph{ultra-sparse}, i.e., we can have an emulator with $n+o(n)$ edges and $β= \left(\frac{\log {\log n}}{ε}\right)^{\log {\log n}(1+o(1))}$. We also devise a distributed deterministic algorithm in the CONGEST model that builds these emulators in low polynomial time (i.e., in $O(n^ρ)$ time, for an arbitrarily small constant parameter $ρ>0$). Finally, we also improve the state-of-the-art distributed deterministic \congest-model construction of $(1+ε,β)$-spanners devised in the PODC'19 paper [ElkinM19]. Specifically, the spanners of [ElkinM19] have $O(β\cdot n^{1+1/κ})$ edges, i.e., at their sparsest they employ $ O\left(\frac{\log {\log n}}{ε}\right)^{\log {\log n}}\cdot n$ edges. In this paper, we devise an efficient distributed deterministic CONGEST-model algorithm that builds such spanners with $O(n^{1+1/κ})$ edges for $κ= O\left(\frac{\log n}{\log ^{(3)}n}\right)$. At their sparsest, these spanners employ only $O(n\cdot {\log {\log n}})$ edges.