Welcome to SG. There are a lot of questions in this, I will try to address the most important ones.
1. You can't just plug in a different modem and expect it to work. Your ISP (CenturyLink) has to support the model, be able to push firmware to it, etc. If you'd like to buy/use your own modem, your only option is to ask them what models they support, buy a supported model, then ask them to provision it once it is connected to the line.
2. Most DSL/cable gateways include multiple devices into one: a modem, a NAT router, and a wireless access point. They can all be part of one unit, as in your case, or they can be separate (i.e. you can plug in a wireless access point to the existing setup just for your wireless network, and continue to use their modem).
3. Extenders should not be affected by the firmware on a gateway/modem, it is most likely a configuration issue.
4. The numbers you quoted for "CenturyLink", 8.8.8.8 and 8.8.4.4 are actually Google's DNS servers (used for hostname to IP address resolution).
I would ask your ISP to assist you in fixing the wireless, or replacing your gateway. I would also let them install all firmware updates, as they may be necessary for the equipment to operate properly. I would also ask them for the login/password to your modem/router, as you need them to configure and secure the Wireless network, to forward ports, etc.
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