This paper presents significantly improved deterministic algorithms for some of the key problems in the area of distributed graph algorithms, including network decomposition, hitting sets, and spanners. As the main ingredient in these results, we develop novel randomized distributed algorithms that we can analyze using only pairwise independence, and we can thus derandomize efficiently. As our most prominent end-result, we obtain a deterministic construction for $O(\log n)$-color $O(\log n \cdot \log\log\log n)$-strong diameter network decomposition in $\tilde{O}(\log^3 n)$ rounds. This is the first construction that achieves almost $\log n$ in both parameters, and it improves on a recent line of exciting progress on deterministic distributed network decompositions [Rozhoň, Ghaffari STOC'20; Ghaffari, Grunau, Rozhoň SODA'21; Chang, Ghaffari PODC'21; Elkin, Haeupler, Rozhoň, Grunau FOCS'22].