Getting back to the original question...................
1968 Russia sent two spacecraft, Zond 5 and Zond 6, on escape trajectories which sent them around the Moon and back to Earth. Zond 5 (at least) managed to return some of its cargo alive. On board were small turtles, flies and mealworms plus some plant specimens. In time the Zond was intended to be a manned spacecraft, but after the flight of Apollo 8 in December 1968 there was no political advantage to Russia in continuing the programme.
Given that the American plan to get men on the Moon required the astronauts to actually pilot the Apollo LM down to the surface, NASA never had any requirement for animals on the Moon. They did inadvertently send some bacteria to the surface on Surveyor III, however. They were still alive when samples were brought back by Apollo 12.