You don't get to decide what the set of integers is, that was decided a long time ago. Everyone here, except you, seems to have a good idea of what the integers are.
If you want to talk about a set that is similar to the set of integers but not actually the same then you have to call that set something other than the integers, otherwise you're just babbling.
If you actually do want to talk about integers then start with a well known definition, something like: the smallest inductive subset of R, closed under negation. Once you settle on a standard definition the rest of the questions can be dealt with. If you can't settle on a standard definition then you simply don't know what the integers are. Sorry.