So... if I have a wallet without a password and create one or change the password, why does it change the master private HD key?
In order to protect future keys in case you were compromised. If an attacker somehow got your wallet at some time, they would be able to access all of your private keys generated previously and in the future. By changing the password, you then make it impossible for someone to be able to steal all of your Bitcoin in the future.
The same is done with non-HD wallets. When you change the password, the look-ahead keypool is refilled to prevent people from being able to steal future Bitcoin.
Given it had already derived private keys and addresses that might already been used, what happen to them?
Are they added as independent keys? So if I don't back them up, I wouldn't have access to funds, correct?
Yes. Those keys are marked as used once the address is used (either requested a new address or an address received Bitcoin). Those are kept individually after a password change.