This paper investigates the application of cooperative non-orthogonal multiple access (NOMA) to a two-tier wireless caching network, where the cache servers and the users are spatially randomly located and the users request cacheable contents that are pushed in advance from a base station to the cache servers. By integrating cooperative NOMA into wireless caching, a popularity-oriented cooperative NOMA pushing (POC-NOMA-P) strategy is proposed for content pushing, while a dynamic cooperative NOMA delivery (DC-NOMA-D) strategy is proposed for content delivery. A new analytical framework is designed to comprehensively evaluate the performance of the proposed strategies. Particularly, unlike existing works that purely focused on the analysis of cache hit probability (CHP) and outage probability, our analytical framework further introduces content diversity gain and delivery diversity gain to characterize the performance gain achieved by the use of cooperative NOMA. Moreover, the tradeoff analysis between performance and energy consumption is also included in our analytical framework. To ensure the analysis accuracy and mathematical tractability, a new tight approximation is developed for the cumulative distribution function of channel gain between two spatially random nodes, with which theoretic results are approximately derived for various performance metrics of our analytical framework. Our analytical results demonstrate that compared with the non-cooperative counterparts, the proposed strategies are guaranteed to achieve higher CHP/lower delivery outage probability and largely reduce the energy consumption incurred by content pushing/delivery. Simulations verify the accuracy of our derived approximations for various performance metrics and demonstrate the superiority of POC-NOMA-P and DC-NOMA-D over existing strategies.