The (public) address linked to a private key can be changed multiple times. However the private key remains the same. This feature I believe is to prevent transactions made from the same address being linked.
No. Your private key will have only one address. You can't get more than one address with one private key. You can create as much as address you want- but all belongs to your wallet; not belong to a single private key.
For instance- in Electrum, you may have 10 receiving addresses; these 10 addresses are generated from 10 private keys. If you generate one more address, it means one more private key.
To have all these addresses backed up without storing private key, you can use the seed key. But as I said; 1 private key = 1 address only and it never get changed.