Question [1]: How did the universe (宇宙) and life (生命) come into existence?
Answer: According to the Buddha’s teachings, due to the natural law of impermanence (i.e. 无常: constant change), the universe (and all in it) rise and fall (生灭) naturally (i.e. not supernaturally) through the cycle of expansion and contraction, which might be similar to the Big Bang and Big Crunch. For more on this, see the chapter on science in https://purelanders.com/new.
Just as the universe’s formation and deformation is cyclical, so do humans, (along with sentient beings of the other five realms) rise and fall in terms of physical-spiritual ‘evolution-devolution’. See https://thedailyenlightenment.com/2010/03/how-were-the-%e2%80%98first%e2%80%99-humans-%e2%80%98created%e2%80%99-part-1-sutta-summary for more on this.
As beings change, they rise and fall in the realms karmically. For more on karma (业力) and the six realms (六道), see their corresponding chapters in https://purelanders.com/new. However, all beings can break free from rebirth (轮回) to attain liberation (解脱); enlightenment (觉悟), as all have Buddha-nature (佛性), the potential for Buddhahood.
Question [2]: When did the universe and life come into being?
Answer: Since they rise and fall cyclically, there is no fixed or single point of origin in time and space.
Question [3]: Why do the universe and life exist?
Answer: Just as there are no fixed purposes for the sun to rise and set, and for the clouds to form and rain in nature, there are no fixed reasons for the universe and life. Thus, we are free to choose how to perceive what both are for. However, the Buddha teaches us how to perceive, as aligned with reality (i.e. realise the truth: 实相). This is perfecting of wisdom (智慧) about all, as attained with Buddhahood (佛果).
With Buddhahood, there is breaking free from the cycle of (re)birth, ageing, sickness and death (生老病死), thus attaining True Happiness. The Buddha also encourages us to to guide one another towards Buddhahood. This is perfecting of compassion (慈悲) for all. In this sense, the ultimately most fruitful purpose in the universe is to progress with one and all (or more) towards Buddhahood.