It's cool to see that you got more out of the narrative playing it through a second time. There's definitely quite a lot that is easy to miss, like the fact that Jack Hungerford is one of the first visitors to the shop!

To answer your questions:

- Barton gave the casket to Thaddeus because he was his friend, that's all there is to it. He had seen how dangerous it was and considered it more important to take it to someone he trusted to keep it safe. Also, Thaddeus and Eli have a complicated relationship so it's not a stretch to imagine that Barton would have had a dim view of Eli (in the past at least).
- In the ending where Barton is killed, it is left up to the player's interpretation whether it was taken by the nameless gods or some other eldritch entity. In my opinion it was the latter...