Aerial relays have been regarded as an alternative and promising solution to extend and improve satellite-terrestrial communications, as the probability of line-of-sight transmissions increases compared with adopting terrestrial relays. In this paper, a cooperative satellite-aerial-terrestrial system including a satellite transmitter (S), a group of terrestrial receivers (D), and an aerial relay (R) is considered. Specifically, considering the randomness of S and D and employing stochastic geometry, the coverage probability of R-D links in non-interference and interference scenarios is studied, and the outage per- formance of S-R link is investigated by deriving an approximated expression for the outage probability. Moreover, an optimization problem in terms of the transmit power and the transmission time over S-R and R-D links is formulated and solved to obtain the optimal end-to-end energy efficiency for the considered system. Finally, some numerical results are provided to validate our proposed analysis models, as well as to study the optimal energy efficiency performance of the considered system.