Short answers, pulled from the story.
The Gaia hypothesis proposes that living organisms interact with their inorganic surroundings to form a self-regulating system that maintains conditions for life on Earth. It was formulated by chemist James Lovelock and co-developed by microbiologist Lynn Margulis in the 1970s.
Lovelock named the hypothesis after Gaia, the primordial Greek deity who personified the Earth, on the suggestion of his neighbor, the novelist William Golding. Golding's choice drew on Gea, an alternative spelling used as a prefix in geology, geophysics, and geochemistry. Golding later referenced the name in his Nobel Prize acceptance lecture.
Daisyworld is a mathematical model developed by James Lovelock and Andrew Watson to show that planetary temperature regulation can emerge from ecological competition rather than conscious cooperation. In the model, black and white daisies compete, and their relative growth rates create a negative feedback loop that stabilizes the planet's temperature even as the Sun's energy output changes.
Margulis defined Gaia as "the series of interacting ecosystems that compose a single huge ecosystem at the Earth's surface," and insisted it is "not an organism" but "an emergent property of interaction among organisms." She dedicated the last chapter of her book The Symbiotic Planet to Gaia.
Critics including Ford Doolittle, Richard Dawkins, and Stephen Jay Gould argued the hypothesis lacked a plausible evolutionary mechanism, since organisms acting in concert to regulate the planet would require foresight that natural selection does not produce. A 2013 book-length evaluation by Toby Tyrrell concluded that the Gaia hypothesis is "not an accurate picture of how our world works" in its stronger forms.
The Medea hypothesis, proposed in 2009, argues that life has highly detrimental and even biocidal impacts on planetary conditions, directly opposing the Gaia hypothesis. It points to events like the Permian-Triassic extinction, roughly 250 million years ago, as evidence that life and Earth's systems can enter self-destructive positive feedback loops leading to mass extinction.