The eternal universe

This (i.e. whether "the cosmos is eternal") is one of the famously undeclared topics -- "undeclared" meaning that the Tathagata wouldn't answer this question; and "famously" both because the question keeps being asked, and because the "parable of the arrow" is famous -- see Cula-Malunkyovada Sutta (MN 63)

I become very nervous when answering questions in a Buddhist forum but here's my take.

In respect of the world of phenomena there are no such 'eternalist' schools of thought. The whole idea is the impermanence of phenomena. If they are not impermanent then the teachings become incoherent.

But clearly the 'Universe' or 'world-as-a-whole' must be timeless or eternal for otherwise we wouldn't seem to be here. Thus there would be at least two answers, one conventional and one ultimate, and this may be why the Buddha would not give a straight answer to this question. It is very difficult to untangle the words unless we already have a grasp of what 'conventional' and 'ultimate' mean here.

The clue would lie in Nagarjuna's proof that nothing really exists or ever really happens. If so, then our idea of an eternal universe must leave out all that exists or happens. Only what is left over would transcend time and space.

So the 'All' as defined by the Buddha in Sankha's answer above would not be eternal, but this would not include what is 'beyond range', i.e. beyond the reach of conceptual thought and sensory empiricism. Even if we know this we cannot describe the situation to someone else. Lao Tsu explains 'Tao that is eternal cannot be spoken'.

Universe is a concept. Concepts are permanent but they have no footing in reality. They exist only in the imagination. If by universe you mean the 'All', this is what the Buddha said about the 'All':

Sabba Sutta: The All (SN 35.23)

"Monks, I will teach you the All. Listen & pay close attention. I will speak."

"As you say, lord," the monks responded.

The Blessed One said, "What is the All? Simply the eye & forms, ear & sounds, nose & aromas, tongue & flavors, body & tactile sensations, intellect & ideas. This, monks, is called the All. [1] Anyone who would say, 'Repudiating this All, I will describe another,' if questioned on what exactly might be the grounds for his statement, would be unable to explain, and furthermore, would be put to grief. Why? Because it lies beyond range."

Is there such a school (in Buddhism, Taoism, Hinduism), which would teach that the Universe exists forever, is never born and never dies? Some names, links, concepts would be fine...

Actually that's just a different way of saying about the same thing. It's exactly because the universe's subjected to endless cycles of birth and death, like all conditioned phenomena, that it's also equally valid to say that it's never really "born" and never really "dies".

See for instance this sutta (DN 27) somewhere in the middle

There comes a time when, Vāseṭṭha, after a very long period has passed, this cosmos contracts. As the cosmos contracts, sentient beings are mostly headed for the realm of streaming radiance. There they are mind-made, feeding on rapture, self-luminous, moving through the sky, steadily glorious, and they remain like that for a very long time.

There comes a time when, after a very long period has passed, this cosmos expands. As the cosmos expands, sentient beings mostly pass away from that group of radiant deities and come back to this realm. Here they are mind-made, feeding on rapture, self-luminous, moving through the sky, steadily glorious, and they remain like that for a very long time.

So for me this means this is repeating with no discernable beginning nor discernable end. The discurse becomes then more to the details:

2. Solid Nectar Appears

But the single mass of water at that time was utterly dark. The moon and sun were not found, nor were stars and constellations, day and night, months and fortnights, years and seasons, or male and female. Beings were simply known as ‘beings’. After a very long period had passed, solid nectar curdled in the water. It appeared just like the curd on top of hot milk as it cools. It was beautiful, fragrant, and delicious, like ghee or butter. And it was as sweet as pure manuka honey. Now, one of those beings was reckless. Thinking, ‘Oh my, what might this be?’ they tasted the solid nectar with their finger. They enjoyed it, and craving was born in them. And other beings, following that being’s example, tasted solid nectar with their fingers. They too enjoyed it, and craving was born in them.

and so on, comparable to old egytian, sumerian and what you want legends. Just read the full sutta, I find it very interesting.