Outside of evolution, what were the leading scientific schools of thought that Charles Darwin contented with when he published his evolution theory as way of natural selection in 1859?
There were many (more or less) non-theological theories of how life had developed before Darwin, starting at the ancient greeks. Many theories included spontaneous generation but also aspects of modification by descent of existing species (i.e. evolutionary change), but most were not that well developed and complete thought. However, one of the more complete evolutionary theories that pre-dated Darwin was Lamarckism, which claimed that individuals pass on traits that they have acquired during their life time. So the basic idea was that individuals respond and adapt to the environmental conditions they experience, and that these traits are then transferred to their offspring. This is one example of a non-theological theory that pre-dated Darwin, but I cannot really say if it was the leading one. I do know that Lamarck's ideas were relatively ignored during his own lifetime.
Darwin included some aspects of these ideas in his "Origin of Species", but challanged Lamarckism when it came to the main processes of evolution. A weak interpretation of Lamarckism has actually seen a revival after the discovery that epigenetics can be transferred from parents to offspring, see e.g. Heard & Martienssen (2014) and Szyf (2014).