We investigate the long-run behavior of single-server queues with Hawkes arrivals and general service distributions and related optimization problems. In detail, utilizing novel coupling techniques, we establish finite moment bounds for the stationary distribution of the workload and busy period processes. In addition, we are able to show that, those queueing processes converge exponentially fast to their stationary distribution. Based on these theoretic results, we develop an efficient numerical algorithm to solve the optimal staffing problem for the Hawkes queues in a data-driven manner. Numerical results indicate a sharp difference in staffing for Hawkes queues, compared to the classic GI/GI/1 model, especially in the heavy-traffic regime.

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