Only sheer speculation is possible here, since our knowledge of the circumstances of the rise of polytheism, especially, is nearly non-existent.
1) What were the central reasons that polytheistic religions developed?
I can conceive of four trajectories that seem likely to me.
1. A natural progression from a more animistic perspective to something more like a pantheon. In theory, the difference between a spirit and a god is one of authority; does this entity have natural authority over a domain of life, or are they responsible only for their own choices within it? In practice, the difference between a very powerful spirit and a minor god is not necessarily obvious. The natural role of Amaterasu in Japanese tradition, for instance, is obviously god-like in nature, and she is often called the "sun-God" of Shinto by outsiders, even though "kami" is best translated as spirit and Shinto itself is generally considered Animist. And in many traditional pantheons, there are gods/theoi/deva whose roles are not clearly defined nor their power over human lives seemingly great, and look to all intents and purposes like personifications of everyday life not unlike those found in an animist culture. This leads to etic confusion often, with the same entity translated into other languages as "god" or "spirit" depending on the biases and expectations of the translator.
2. It would also be easy to see polytheism developing as a response to the locality of godly domains; if the neighbors honor a different god than yourself, imagining a sort of council of Gods with divided authorities as per Palestine during the Biblical era is a more mild alternative to insisting that only of of those gods
exists.
3. You do not mention binary theism here, but as a subvariant of polytheism it is interesting. I see it as a response to the seeming paradox of human life, forever filled with abundance and suffering simultaneously.
4. Polytheism would be the most natural course for a system of ancestor worship to follow; if veneration of those passed on is the purpose of one's devotions, it is inherently obvious that the dead are many in number, even if only the totem of a clan grouping is the target there are always many clans.
2) Did monotheism often follow from polytheism, or is this a false premise?
By numbers, most monotheists belong to one of three faiths, all of which have a shared background in the polytheistic context of the ancient Near East. In a numbers sense, this seems clearly correct.
We shouldn't assume a polytheistic background for all monotheistic faiths, though. Monotheism of a sort was commonplace in the ancestral US Southeast, for instance, despite the absence of anything strongly resembling polytheism in the same area. Moreover, even relative to animism, anyone would tell you that the Great Spirit is not, substantially or connotationally, a spirit in the sense that first man, coyote, or spider are. This was something different.
3) What were the central reasons that monotheistic religions developed?
A more interesting question to my mind, as several monotheistic religions have arisen as the result of revitalization movements in recent memory. And the answer, almost to a one, is personal revelation. Monotheistic gods reveal themselves to a chosen prophet, whose authority extends over a small group of believers as a result, eventually elaborating into a religious structure should the group remain solvent following that individual's death (not a given). Someone here might correct me (?) but I have never actually heard of a religious movement that started with personal revelation from multiple deities at once.
Exception: Plato, who seems to have just
reasoned his way into monotheism.