Pattern-avoiding permutations are a central object of study in both combinatorics and theoretical computer science. In this paper we design a data structure that can store any size-$n$ permutation $τ$ that avoids an arbitrary (and unknown) fixed pattern $π$ in the asymptotically optimal $O(n \lg{s_π})$ bits, where $s_π$ is the Stanley-Wilf limit of $π$. Our data structure supports $τ(i)$ and $τ^{-1}(i)$ queries in $O(1)$ time, sidestepping the lower bound of Golynski (SODA 2009) that holds for general permutations. Comparable results were previously known only in more restricted cases, e.g., when $τ$ is separable, which means avoiding the patterns 2413 and 3142. We also extend our data structure to support more complex geometric queries on pattern-avoiding permutations (or planar point sets) such as rectangle range counting in $O(\lg\lg{n})$ time. This result circumvents the lower bound of $Ω{(\lg{n}/\lg\lg{n})}$ by Pătraşcu (STOC 2007) that holds in the general case. For bounded treewidth permutation classes (which include the above-mentioned separable class), we further reduce the space overhead to a lower order additive term, making our data structure succinct. This extends and improves results of Chakraborty et al. (ISAAC 2024) that were obtained for separable permutations via different techniques. All our data structures can be constructed in linear time.