I see you've been given a lot of misinformation and are still looking for answers. Let me clear this up for you, and show you how you already determined what the problem is. First of all, "DNS Not Responding" is a generic error message. It doesn't actually mean that the DNS service of your computer isn't working, or that the DNS servers you use are down. It simply means "you're not connected to the internet for some reason", but your computer cant determine exactly what that reason is, so it just generates the "DNS Not Responding" error as a fallback.
Now, the reason you're not connected to the internet could be any number of things, such as your computer, your router, your modem, your internet provider, or something else entirely. Its important to narrow down this list and determine where the error exactly is. And you've done this. You removed your router from the equation, plugged your computer into the modem, and still experienced the error. This means the router was not at fault, its still an issue with either your computer, the modem, or your internet provider. You called your ISP and they had you power cycle the modem, which unfortunately only fixed the issue temporarily. But the fact that it fixed it at all means you're on to something! Its the modem! You rebooted the modem and got internet service back. But it crashed again. So you reboot the modem again, and it comes back again. Rinse Repeat. If the modem keeps disconnecting your internet connection, its probably time to replace the modem. If you currently rent it from your ISP, demand that they bring a new one to you. If you own the modem, you may want to look into replacing it yourself. I know this can be expensive, so you may want to have the ISP troubleshoot things on their end before dropping money on new equipment.
Now, why the "DNS Not Responding" error is generated. Think about it - when you go to a website such as www.yahoo.com, your computer makes an outbound request to your DNS server to look up the ip address for Yahoo. And it waits for a response. And it waits. And it waits, but you never get a response. It never gets a response because your modem has already dropped your internet connection. You're offline! Of course the DNS server doesn't respond to an offline computer. It cant! So, you're browser generates that error message and leads you down a path thinking that it is a DNS issue when in reality its not.
TL:DR - Not a DNS problem. You already found the problem. Replace the modem and/or have the ISP fix it themselves.