Only because you arbitrarily assign one half to be the ciphertext and the other the key, you could swap them and the result would be the same.
No. Then you would be sending your message in the clear and encrypting the pad. The point of OTP is that you can pre-share the pad, then later use it to exchange messages without the message being read by intercepting parties.
For the most part it's just as difficult to send both halves as one whole
You don't send them as a whole. You pre-share the pad (eg: tell your friend to use the prime-numbered pages of a specific edition of the bible), then you can freely send messages you've encrypted using that pad, up until the point when you've used up the pad.