The necklace obviously still had an impact on Frodo, so Sam had to carry him. The chicken takes the place of Frodo, so both Frodo and Sam can carry the chicken.

Sauron wore it in the great battle, and he was by all accounts visible, otherwise how could Isildor cut the finger off? That would be a very lucky swing if Sauron was invisible.

No, I think its power depends on the wearer. Frodo didn't want to be seen, so it made him invisible to everyone but Sauron, who understands its power.

I've never read anything beyond the Hobbit, but I believe the Silmarillion goes in to more detail on how the ring would affect other races should they wear it. I particularly recall someone mentioning how the ring would affect gandalf, but again this is all secondhand knowledge.

Wearing the One Ring does make others invisible because one thing it does is take you into the unseen world (where the wraiths are visible and not just cloaked) and basically most people think you'd have to have the willpower and wear the ring for long enough to drag yourself back into being seen.