This paper investigates a design framework for sparse fluid antenna systems (FAS) enabling high-performance direction-of-arrival (DOA) estimation, particularly in challenging millimeter-wave (mmWave) environments. By ingeniously harnessing the mobility of fluid antenna (FA) elements, the proposed architectures achieve an extended range of spatial degrees of freedom (DoF) compared to conventional fixed-position antenna (FPA) arrays. This innovation not only facilitates the seamless application of super-resolution DOA estimators but also enables robust DOA estimation, accurately localizing more sources than the number of physical antenna elements. We introduce two bespoke FA array structures and mobility strategies tailored to scenarios with aligned and misaligned received signals, respectively, demonstrating a hardware-driven approach to overcoming complexities typically addressed by intricate algorithms. A key contribution is a light-of-sight (LoS)-centric, closed-form DOA estimator, which first employs an eigenvalue-ratio test for precise LoS path number detection, followed by a polynomial root-finding procedure. This method distinctly showcases the unique advantages of FAS by simplifying the estimation process while enhancing accuracy. Numerical results compellingly verify that the proposed FA array designs and estimation techniques yield an extended DoF range, deliver superior DOA accuracy, and maintain robustness across diverse signal conditions.