The eternal universe As far as I know, in Buddhism and Hinduism, the universe has birth and death, like the time of life. However, in Buddhism, as well as other Eastern religions, there are a lot of schools and movements. 

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...
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)
&quot;Monks, I will teach you the All. Listen &amp; pay close attention. I will
speak.&quot;
&quot;As you say, lord,&quot; the monks responded.
The Blessed One said, &quot;What is the All? Simply the eye &amp; forms, ear &amp;
sounds, nose &amp; aromas, tongue &amp; flavors, body &amp; tactile sensations,
intellect &amp; 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.&quot;