Why are there polytheistic, monotheistic and even atheistic schools of Hinduism?

I don't think this question has an answer in the way "What arguments did the Moslem League put forward to advance the partition of India?" has an answer. Why are there different schools of Hinduism? shrug Why are there so many different schools of Christianity? Because people are different and religious scriptures leave themselves open to interpretation.

I don't know how much influence these schools have over your average lay Hindu, but it is strange that such contradictory philosophies are supposedly part of the same religion.

First, it's unhelpful to think of Hinduism as a single religion in this context like Christianity or Islam. It's more accurate to think of Hinduism as religious complex like Abrahamic faiths, and from that angle, the broad diversity of tradition is a bit more understandable.

So how did these different schools develop

This is a big question. Far too big for me (who specializes in Buddhism) to even begin to answer. I will say that there are many many volumes that try to untangle the different branches, schools, an sects of Hinduism from many different angles. Me, being historically minded, started with John Keay's India: A History though there's less religious information in it than I would have liked. The Hindus: An Alternative History by Wendy Doniger is interesting, though it has some problems with interpretation and the author leans a bit too much on a modern framework IMO. It might be easy to start with one of these to give yourself a framework of reference and then branch out using their bibliographies.

Another text which is religious in basis, but is a good internal view of Hinduism is Dancing with Siva by Sivaya Subramuniyaswami. The latter part of the book includes an overview of the world's philosophical systems, including the four main branches of Hinduism which is a very helpful summary (of course, again we're dealing with a modern religious text which is obviously Saivite, so it's worth keeping that in mind).

All in all, there are four main branches of Hindu religious thought, two major and two minor: Vaishnavite, Saivite, Shaktist, and Smartist. Each one (except for Smarti) is named after their primary deity (Vishnu, Siva, and Shakti). Smarti is named after "smriti" whcih are the post-Vedic literature attributed to human authors. It has been said that the ideal Hindu has a public Vaishnavite practice, a private Saivite practice, and a secret Shakti practice.

The basis of Vaishnavism is a series of texts around the first millennium, including the Bhavagata Purana, discussed a bit in this thread, that focus Hindu thought and worship around Vishnu, and particularly his avatar Krishna. Vaishnavism is most well-known for its focus on the idea of avatars being the basis for God's role on earth, and that deities take shape now and again. Vishnu in particular, takes shape to restore order and religion to the world in times of apostasy. Vaishnavism is the most public and recognizable branch of Hinduism because it is so popular and a lot of the festivals, ideas, and concepts we associate with Hinduism are based in Vaishnavism (similar to how, say, my Indian Professor keeps referencing "Rome" as a holy land for all Christians, despite the fact that my Protestant upbringing would have found that to be heretical). The International Society for Krishna Consciousness (ISKCON), aka Hare Krishnas, are Vaishnavite.

Saivism is rather close to Buddhism. I say this because later forms of Buddhism (the Buddhism we see in Tibet in particular) grew together with the Tantric tradition mostly out of Kashmir. Tantric literature, which utilizes violent and sexual imagery (though from experience, 90% of it features how to chant mantras correctly, but this isn't nearly as scandalous). Saivite philosophy and theology bleeds a lot into what we think of as mysticism in the west (God is one and all is God sort of New Age stuff) because that's all there to some degree in Tantric literature. Though to go any further would require more base knowledge in Vedic or Buddhist philosophy.

Shakti (broadly speaking) is the form of Hinduism that focuses on the divine feminine and goddess imagery. Smriti thought is more or less a blend of all of the above as it seeks to reconcile the thought of the polytheistic Vaishnavites*, the pantheistic Saivites*, and the yonic Shaktists*.

*This is by no means a comprehensive summary of Hindu tradition and thought. I'm trying to give you a place to start looking and searching, but just dipping your toes in the water of Hinduism here is itself a daunting endeavor. Each school of thought listed above has subschools, primary, secondary, and tertiary deities of importance, plethoras of ideas, not all of them contradictory, regarding whether God is one with many faces, God is one but fractured, there are many Gods in a complex, or whether everything is God and reality is an illusion.

why are they all considered part of Hinduism while the Buddhists and Jains (who are also related) are not?

There are texts in each tradition that are regarded as unique and are not considered to be in the others. But as you've indicated, each tradition in Hinduism holds the Vedas to be central to their understanding of God and the universe. (And to another degree Varna, or caste, is also considered to be sacred. One Indian professor I had refused to call it "Hinduism," because it was the Brahmins who held authority, so it should really be called "Brahmanism.")

They are all a part of Hinduism because the defining features that make up Hinduism (that they hold the Vedas and Brahmin sacred, that they worship the Divine in some form, that they hold India as their sacred land) are all present. Buddhism (and Jainism, if I understand it correctly) are different because they categorically reject the authority of the Vedas, upon which Hindu thought and philosophy is based. By rejecting the Vedas, Buddhists are open to ideas that are antithetical to Hindu thought. (I.e. one of the major sticking points between Hinduism and Buddhism is that Hindus believe in atman, the self or soul. Whereas Buddhists reject this and insist in anatman, the doctrine of no self or no soul. Recall that Vaishnavites believe in multiple avatars of God, this is the piece that moves between physical bodies, the atman. Saivites, however, would say that the atman that inhabits you, and me, and all things, is all a part of the same deity.)

One final footnote: Buddhism and Jainism are legally speaking considered subcategories of Hinduism in the Indian constitution. No, it makes no sense considering that Buddhists and Jains reject the authority of Hindu philosophy, but there it is.