I honestly don't think you can improve upon the title of the story, although of the variations, I vastly prefer "Lady with the lap dog" to "Lady with the Pet Dog"

Personally, I would call it either "Escape" or "Paradox." "Escape" would refer to both the vacation setting and the way Gurov works to get close to the woman, and then pulls away. It would also address the woman and her lack of knowledge of her own life, which is a kind of escape. "Paradox" would be for similar reasons; they are on vacation, but don't seem to be happy. They seek closeness, but find isolation, etc.

hmm...that's an interesting question as Chekhov has a way of giving the most wonderful titles. I think personally i would do something like, "The inferior race." I chose that because its an expression that Gurov uses to refer to woman in the beginning of the story, but his opinion is so changed, that to look at that title at the end of the story one knows the transformation that Gurov went through. I do think, however, that Chekhov would abhor my title, but i like it.