[Hookbill the Goomba]

I think it’s well documented that Tolkien was not 'a cat person', but I'm not convinced that The Ring was a direct parody. As Tolkien says in the forward, "Many people mistake allegory for applicability". I think this is a case of applicability; The Ring can have many meanings to the reader and writer (I'm trying not to get into the canonicity argument here). But, yes, The Ring did have many 'cat-like' qualities as you said.
I cannot say that I think The Ring stayed with Sauron as it suited it. I think The Ring genuinely believed that Sauron was the ultimate master; going to others was just a way of trying to get back to him, through the weaker minded. I don't think The Ring would even let someone who wielded it overthrow Sauron, in a way; Sauron had The Ring under his spell... Well, that's how I read it...