The extent of the Lion's cowardice seems to differ a bit from one story to another. Baum usually presented him as afraid of just about everything, but never letting that stop him from acting bravely when necessary. Ruth Plumly Thompson kept that characterization, but also made him more prone to panicking and fainting on occasion. One of my favorite portrayals of the Lion is in the McGraws' Merry Go Round, which gives him some sarcastic wit in addition to his other character traits. I think that, of Dorothy's three Ozian companions, the Lion is the one most misrepresented in the MGM movie. They keep the general idea of his being a coward who steps up when necessary, but he's played in a much more exaggerated and undignified fashion than the Lion of the books. That's not to say that I don't find Bert Lahr's performance amusing, but I think it's led to some misunderstandings about the big cat of the original books.
The Cowardly Lion's early history is never really explored that fully, but Dorothy finds him in a forest in the Munchkin Country, and he becomes ruler of another forest in the Quadling Country when he kills a giant spider that had been terrorizing and devouring the animals there. From Ozma on, however, we usually see the Lion in the Emerald City, serving as Ozma's bodyguard and chariot-puller along with his friend, the Hungry Tiger. Onyx Madden's Mysterious Chronicles, Fred Otto's Lost Emeralds, and Edward Einhorn's "Ozma Sees Herself" all give somewhat different accounts of how the Queen met the two famous felines.