The fact that your network is being identified as public, in that it is not receiving proper IP addressing from the DHCP server on your network is the actual reason you do not have internet access. It is not two problems it is a single common problem.
I wouldn't suspect that you would need a disk for the ARRIS modem, I have the same or at least similar device from Comcast (Xfinity) and it doesn't require any drivers. The network adapter in your computer however does need a driver and dell will likely have one that should work, but you can also attempt to obtain a more current driver from the device manufacturer's website.
How do I update a driver?
Regarding your suspicion concerning the Bonjour service. You just need to double check and make sure that it is no longer enabled in the services console. A good indicator is usually just to launch
Driver Install - Manual Method
- Download the driver and extract it to a folder.
- Open the Device Manager.
- Right click the device you wish to change the drivers for. You can select Properties > Driver > Driver Details to check you have the right device.
- Choose Update Driver Software.
- Select Browse My Computer For Driver Software.
- Select Let me pick from a list of device drivers on my computer.
- Select Have disk...
- Select Browse..
- Navigate to the extracted folder and double click on the .inf file
- Hit OK, then Next and the driver(s) will then begin to install.
- After it has installed, GoTo C:\Windows\System32\Drivers. Locate the driver you've just updated, right click it and check the dates under its properties.
Task Manager and see if you see something called mDNSresponder.exe running. If so then either Bonjour or one of its' several cousins are still present.
Now regarding your network. You do have a network. When you plug your computer into a network appliance, it sort of just happens, you have a network. Yours is being identified as Public, because it is not receiving proper IP addressing and does not have a default gateway and as a result you are going to be stuck on Public until you can resolve this and then you'll be prompted as to how to define it as Private (Home or Work)