two ring questions

So, some of this is answered in the greater lore Tolkien left, other parts might be muddy because they were never super-defined and/or might differ in the films.

Question 1: I think the spirit world and the physical world both exist same time/space, but mortals are not privy to the spirit world. Elves, wizards, balrogs, etc., can access the spirit world. Legolas and other elves can practically see in the dark - probably because they have some spirit-sight-sense that we don't have (that's not in the lore, I just used that as a possible "for instance"). Some beings seem to have become partly detatched from the physical world the way that men and hobbits can't tap into the spirit world. The Ringwraiths have poor eyesight because they can't really see the physical world anymore, but when somebody enters the spirit world (puts on the Ring), they can see that sucker plain as day.

So, can things exist in one world, not the other, and be brought back-and-forth? Seems like it. It might be hard to do, though. The object passing you're referencing is, I believe, film stuff, although it could be in the Unfinished Tales or something like that. But I'd guess it would be possible to detach something from its physical form and wraithify it, but that would be REALLY hard. Sauron winds up spirit-only by the end of the Rings, and even then he's sort of adrift and scattered.

Things are kept nebulous there, and so there aren't 100% rules here.

Question 2: Yes. If Bilbo put the Arkenstone into his jacket, I think it would disappear. But if he was just holding it, it would be visible. In the books, Ring-wearing folks aren't visible by undergarments or smoking pipes they have in their pockets. This suggests that objects in clothing or worn on a person fade out, too. But we do see weaponry. For example, the orcs in the tower where Sam rescues Frodo see Sting hovering and slashing at them and they are terrified, thinking themselves under the attack of some demigod invisible elf lord.