Security of expanded key
Does this leak information about the keys?


It leaks information about the plaintext.  For example, given the three ciphertexts, it's possible to compute the XOR of the second and third plaintexts.  This is often enough to deduce the original plaintexts.


  What if some sort of random mapping where each bit of the key does not encrypt consecutive bits of the next key was thrown into the mix?


To start, instead of trying to guess about how to make the process more complex, I would suggest thinking about how to rederive the attack I mentioned above.  Learning about how to do cryptanalysis is the right place to start to learn how to design cryptography.