Hi there,
I hope I can help.
1) There are arguments for and against using 'the' when talking about Earth. You will see both in use, though many people will argument 'the' isn't necessary. You wouldn't get any funny looks for using either.
2) I can imagine it does seem strange, but if you were to expand the sentence and look at it this way, "No, I haven't (seen your hot wife)", you can see that you're answering the question and confirming the answer.
It would be a common response in other situations too;
"Did you drink all of the milk?" - "No, I didn't".
"Is it Tuesday?" - "No, it isn't."
3) These are interchangeable and your examples are absolutely fine. You may also see "My phone is AN Apple" when it is a well known brand. "Is your laptop a Lenovo?" - "No, it's an Apple". The differences are small enough that you don't need to worry.
Hope I helped!