The rolling of IP addresses is controlled by the DHCP server at the cable headend. Many of them are configured specifically to provide you with the same address as long as the MAC (hardware) address of the connected device is the same, and some provide a "pool" of addresses to a specific cable modem, up to however many addresses are permitted by the specific account settings.
The CMTS (cable modem controller) is usually provisioned to allow more than one device to connect to the cable modem, depending on your account. If you plug in too many different devices, the cable modem will stop learning their MAC addresses and the new ones won't get an IP address (traffic blocked).
Resetting the cable modem will clear the local MAC list, so if you replace old devices with new you can clear the counter, but the CMTS also maintains a list, so simply going into your router, changing its MAC address, and rebooting the modem may work a few times but unlikely to work indefinitely. If you inadvertently pick someone else's MAC address and use that, the CMTS will detect the MAC coming from two different modems and block the new one.
The battery backup on your cable modem will help keep your connection working in a power failure, but doesn't impact your IP address at all. Modem model is irrelevant, as they all adhere to the DOCSIS standard. The DHCP server may flush IP address-to-MAC address mappings occasionally, some providers do this to change your IP address every x days (usually no less than 60). Your IP address will also change if the network subnet serving your area changed, like if the fiber node serving your neighborhood was split into two, reducing the number of homes on a given fiber strand.
