Short answers, pulled from the story.
Cosmogony is any model or narrative concerning the origin of the cosmos or universe. Cosmology studies the universe's structure and existence without necessarily inquiring into its origins. In astronomy the two are treated as distinct fields, though in theology and the humanities the boundary between them blurs.
The Hartle-Hawking state, associated with Stephen Hawking among others, asserts that time did not exist before it emerged alongside the universe. This implies the universe has no beginning, because there was no prior moment in which a beginning could occur.
The standard Big Bang model assumes densities low enough to exclude quantum effects, but as the model is followed back toward time zero, densities rise until general relativity breaks down. The model cannot be extrapolated all the way to time zero, leaving the ultimate origin outside its reach.
The horizon problem refers to the fact that opposite sides of the observable universe are far more uniform than an extrapolated Big Bang can explain. Events on opposite sides of the cosmic horizon could not have exchanged energy in the early universe, so they should not be homogeneous now.
Eridu Genesis is the oldest known creation myth. It describes the world being created out of a primeval sea called the Abzu.
Mircea Eliade and his colleague Charles Long identified five types: creation ex nihilo from nothing by a divine being, earth diver creation where a creature retrieves mud from a primordial ocean, emergence myths where progenitors pass through successive worlds, creation by dismemberment of a primordial being, and creation by splitting or ordering a primordial unity such as a cosmic egg.