Foundation (Apple) [** SPOILERS **]

I see nobody has replied to this yet, so if it's still relevant to you...

The box is called the Prime Radiant. Think of it as a highly advanced version of a mathematician's blackboard in the movies, covered with formulae. It projects a visual representation of the mathematics, that's the "pattern" which you see with the various crises, branches and so on. If they were to zoom in many thousands of times, you would see that the 'pixels' are actually mathematical formulae. Remember, psychohistory is a mathematical assessment of probability. I don't recall that the prime radiant was ever properly explained in the show, but that's what it is. In the books it was really just a combination workspace and projector, that it was a complicated enough machine to contain a consciousness is a deviation for the Apple TV variant.

It was mentioned briefly in S2Ep 2, but the reason the radiant wasn't given to many people was that psychohistory is premised on natural actions. If people knew the maths, then they would be affected by their knowledge of the future and thus immediately undermining that maths. Again, it is a deviation for the show that one Psychohistorian (Gaal) was to be sent to Terminus with a Prime Radiant. In the books, there were no psychohistorians at all. (They were in the Second Foundation). The second foundation's job is to make changes to the mathematics to reflect real-universe deviations from the originally determined probabilities and to take actions which will keep The Plan unfolding as intended. This is, in effect, what Gaal/Hari/Salvor are doing with the "updates to the cube" which you observe.

I'll pass on the Hari respawn as I haven't gotten to an episode which explains it yet. The entire Hari arc is a deviation from the books, where he dies of old age at his desk on Trantor. (And married a historian, not a mathematician).

The Empire diverging from the Empire's plan, you are correct. You are also correct that the Empire didn't care about the Foundation until it started incorporating other planets, this matches with the books (though the details differ somewhat. In the show, the Empire thought the Foundation was destroyed, in the books, the Empire thought the Foundation was just a group of Encyclopaedists archiving knowledge somewhere safe and thus not worth worrying about). In both the book and the show, Bel Riose is sent to bring the expansionist foundation to a heel.

The Vault, which killed the warden (which I thought was overall a bit out of character) is not quite the same as the Prime Radiant, though apparently it does have the ability to hold Hari's consciousness which presumably has access to a radiant anyway. The purpose of it is really to reassure the Foundationers that they are on the correct course and to explain to them just what happened. In the books, the vault had no consciousness, it was just a chamber which played recordings Hari had made when he was alive. The recordings were on a timer. On one occasion, the recordings played when nobody was around, on another it was entirely incorrect. In the show, because the Vault contains Hari's consciousness, it can perform a slightly more interactive role in resolving the crisis, but the crisis itself and its resolution are still primarily brought about by the inevitability of psychohistory. The vault will only open for a crisis.

It is a kind of shrine to the Foundation, but not in the religious sense. More like The Alamo or the GPO. The religious aspect you see with the two Brothers is a mechanism the Foundation used to spread its influence, the Foundation itself (and even the Brothers) have faith in Psychohistory and Seldon as a science, but not in the deity context. The religious aspect was actually used in the books to solve the first crisis, but again, TV show deviations.

Hope that helps.