I rent a cable modem from my ISP that has its own Wi-Fi capability. Does my ISP have the ability to track various settings on their modem, such as wireless network SSIDs, etc?
Also, if I were to add an additional wireless access point, would my ISP have any visibility to the access point’s settings?
Yes,
Modern routers in their firmware have a setting to support CWMP (Customer-Premises Equipment WAN Management Protocol) for remote management. This is also called TR-069 by its name on Technical Report 069 as it first introduced. This capability is also called zero-touch configuration and most providers are now using for remote-configure your router.
I rent a cable modem from my ISP that has its own wifi capability. Does my ISP have the ability to track various settings on their modem, such as wireless network SSIDs, etc?
Yes, see @ebal's answer.
Also, if I were to add an additional wireless access point, would my ISP have any visibility to the access point's settings?
If you added a second access point, then no. You should set a password on it to be sure(on the AP itself). However, if you added another SSID to your rented AP ( making the one hardware device appear as two wireless networks), then yes they would have access to those settings, as well as any others on their box.
If you use a device provided by someone you should take into account that they can customize the firmware as they wish - that includes leaving an SSH server with their key authorized to access the root account, so they can remote in at any time and do whatever they want on the router.
Even if they do not have direct remote access the firmware may be designed to periodically check for updates from the ISP's server, so they can add the remote access functionality at any time.
Now, this doesn't usually mean it will be used maliciously. I do it as well on the equipment I provide, and I feel it is fine as long as the customer is fully aware of it. This allows for quick fixes and diagnostics before the customer even realizes there was a problem in the first place. But you have to understand that the level of control your ISP can have is way beyond seeing what SSID and wireless key you set. With full root access it is possible to see and tamper with the traffic passing through the device.
Whether you trust that router depends on whether you trust your ISP and how their systems are secured (if an attacker steals the credentials used to log into your router or compromises the remote management/update infrastructure he will have the same level of control the ISP has).
Finally, with most ISP-provided routers, I would be more scared about the lack of updates and security built-into the firmware rather than the remote access functionality. Remote access is bad, but it's likely there are tons of other vulnerabilities that give full root access to an attacker no matter whether ISP-provided remote access is in place or not.
For your second question, no - your ISP will not be able to see your own equipment's settings unless they know your administrative password (and will login over the web interface just like you do when configuring the device). They could, however, still know which SSID you are broadcasting and what encryption type, as they have control over the router and the wireless card and nothing prevents them from running airodump-ng on it. I doubt they will do this (why is knowing your own SSID so important to them?) but they definitely can.
In general, if someone gives you a communication-enabled device and you use it to do a thing, there's a very good chance that whoever gave it to you can use some kind of remote management to see exactly what kind of thing you're doing. Since your wireless network is managed by a device from your ISP, and that device obviously can communicate with your ISP, they could - and likely have - included features that send info about your WiFi to them.
If you use a device that's not supplied by your ISP - e.g. a separate wireless router - then your ISP can't fiddle around with it (without knowing the password) unless the router was specifically designed to smuggle configuration details out the data stream that passes through the ISP. I personally find that scenario unlikely. If you're still paranoid, you could flash on open-source router firmware like Tomato.
Of course, if your wireless router is broadcasting its SSID, a particularly sneaky ISP could capture it with a listening device placed near your house. That device could be hidden in the box where the cables come into your house, or it could even be embedded in the ISP-supplied cable modem.
