What are the conditions necessary for life to take root, evolve, and flourish? Where might we find these outside of Earth? The solar system?

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What are the conditions necessary for life to take root, evolve, and flourish? Where might we find these outside of Earth? The solar system?
How can we define life?
We may give a reasonable biological definition of life. Life in a system means it is capable of metabolism, growth, reaction to stimuli, reproduction, mutation and reproduction of its mutation.
For life to take root, the surrounding conditions such as temperature, radiation effects from stars, abundance of natural elements, etc., must be supporting. The atmosphere plays a major part in the creation of life and sustaining it.

Life can exist in the strangest inhospitable places on earth. Bacteria or algae are those which apparently choose extreme life conditions. Especially in the deep-seas live so-called "super-thermophilic" (=extremely-heat loving) micro-organisms, these not only exist, but thrive at temperatures even beyond 150 ° C. They inhabit pressurized environments beneath deep-sea hydrothermal vents. At these temperatures you might expect the water to boil, but it doesn't because of the immense pressure. Recently bacteria were discovered which live at an astonishing 169 ° C! It's not only the temperature, lots of organisms at places where no sunlight ever comes, use chemicals like hydrogen sulfite as their energy source. The bacteria in turn sustain larger organisms in the vent community. It is the increasing temperature that limits the depth of life beneath the surface. In the oceanic crust the temperature rises about 15 ° C per kilometer. Thus microbial life extends on average about 7 kilometers below the sea floor. For continental crust the microscopic life should reach almost 4 kilometers into the earth, for the surface temperature is approximately 20 ° C and it rises with 25 ° C per kilometer. However the amount of micro-organisms will vary from place to place. Saying about bacteria, at too cold temperatures they go into some sort of stasis, you could compare it with hibernation, and they do not show any characteristics of life. If the temperatures increase they become active again. At too hot temperatures, however, they are damaged too severely that the damage is irretrievable.
If we want to investigate the possibility of life on other planets we have to figure out what exactly we are looking for. You cannot go marching off looking only for planets that resemble Earth as it is nowadays. We shall have to go back to the formation of the Earth and life's origin, things were very different then...
Earth formed about 4.6 billion years ago (= 4.6 Giga years) as third planet in a series of nine circling around a star, the Sun. The Sun and the planets are made from stellar debris of stars that came to the end of their existence. Everything we see around us is actually recycled stardust. After the formation, Earth was a pockmarked planet of roughly uniform composition and had an early atmosphere of mainly hydrogen. Then ...