I want to know definietly: Is Auri-El and Akatosh that same being ?

I have my own point of view about this, but I want to hear lore experts.

How I see this:

Akatosh as a Dragon of Time is "bigger and elder" than Auriel and Aldun etc. I'm assuming that term "Akatosh" wasn't invented by St. Allessia, but it functioned earlier (Alduin and Galebor refers to God of Time as Akatosh).

Akatosh is not Auri-El nor Alduin, Alkosh etc. But Auri-El, Alduin are Akatosh, at least they are some part of bigger Divine. That sayng Akatosh is supreme to his counterparts present in different cultures.

Alduin said that he is the firstborn of Akatosh - Ok, I understand that he represents one aspect of greater divine and acts autonomiccaly from it. Now I dare say something heretical: What if Auri-El is also born from Akatosh? I mean not that Auri-El=Akatosh, but Akatosh>Auri-El.

So we're talking about Akatosh the original time dragon, not the Alessian Akatosh. I think you have it down pretty well. All of the "time-gods" of the various cultures on Nirn (Alduin, Auriel, Tosh-Raka, Alklosh, etc) are derived from the original Akatosh. But I don't think there's an actual source on it, besides Alduin's case and the book Fire and Darkness (EDIT, not the right book, I don't remember the name), so it's mostly just a theory now.

EDIT: Found the book: The Light and the Dark. Says the spirits are affected to at least some extent by mortal belief. According to that logic, all the different cultural time god's are shed from the original Akatosh due to the power of faith.

Quite simply, they are all one. Akatosh is depicted with several heads, not just because of his creation by Alessia in the merging of pantheons, but also because he is schizophrenic and contradictory by nature. They are different ways of looking at the same end of the axis. Lorkhan, on the other hand, is the other end of that axis, but is also joined on. This reflects the schism and unity and contradiction inherent in the entire universe.

It does indeed get complicated by Auri-El appearing in physical form and leading the elves. But see Talos=Lorkhan: Auriel as a warrior is much the same idea. Transcending to become the gods and displacing them and also fulfilling the place you were always meant to.

The reason gods can appear to be mortals and vice versa is just... one of these things that happens in the messed up Mundus. Pelinal, Morihaus, "avatars" of gods are all et'Ada. As such it's difficult to distinguish them and say that one is "born from" the other. The real connections that are identifiable between gods (besides their unity) are those which are imposed by myth and mortal perception.

The reason why it appears that Auriel and Alduin are part of Akatosh but Akatosh is not fully them (speaking within the rational beliefs of common people) is not exactly clear to me. Is there something in the Imperial psyche which lends itself to accepting contradictory gods? Or is it just something about Aka? Maybe it's Marukhati fiddling that makes him look that way. In which case, he probably always was that way.

You'll also want to reference this rather famous quote from MK on the topic:

"Don't forget that gods can be shaped by the mythopoeic forces of the mantlers-- so Tosh Raka could be an Akaviri avatar of Akatosh with a grudge against his mirror-brother in Cyrodiil.

Just like Akatosh-as-we-usually-know-him could time-scheme against his mirror-brother of the Nords, Alduin, to keep the present kalpa-- perhaps his favorite-- from being eaten.

Notice all the coulds."

Hence the contradictory-ness of Akatosh noted by others above. Shor son of Shor ( http://www.imperial-library.info/content/shor-son-shor-full) is also a good reference to show how the Aedra, in general, can be on multiple sides of a conflict, as they are shaped by mythopoeic forces of mortal belief.

So are they the same being? Best way to answer this I can think of is: Yes, sort of... not anymore.

Auriel, Alduin, etc. are all variations on the same theme. I'm hesitant to call that theme Akatosh, because it is not the same as the imperial god. I believe Aka/Borhamu/whatever-you-want-to-call-it really only existed before there were mortals to alter his shape. He exists now in a thousand self-defeating fragments.

I think it's important to point out that Paarthurnax names the dragon father-god "Borhamu" when speaking in the dragon language, and just translates it as Akatosh because that is the name that would be most familiar to your ears. Not because that entity has more validity.

Yeah, but Al-Du-In mentioned Akatosh, before that name was invented by Allessia, so I assume that this is his True name, and Allessia just sticked it to her diety.

To be fair, the Marhukati did some messed up stuff to the Time Dragon in an attempt to purge Auriel, possibly dragon-breaking Allesia's Akatosh to be the original Time Dragon.

It gets messy.