I want to connect my Galaxy S2 HD LTE SHV-120S Android version 4.0 (rooted) to the internet via a LAN connection. I am sensitive to wifi and network radiation, hence want to make a line connection and keep my phone on airplane mode. BobjGear mentions some tablets have the required settings, but no phones. I have an OTG cable, a USB LAN adapter, and a usb cable. I did not find any ethernet drivers on Google Playstore. Would CyanogenMod or Ubuntu OS have this support? If it is not possible with this phone, which phones can be configured for it?
You'll need a USB-OTG adapter, or equivalently for newer devices with USB-C ports, a USB-C to USB-A adapter, to be able to physically connect a USB Ethernet adapter to your phone. It'll only work, though, if the phone has a driver for the Ethernet adapter you're using. That may vary from phone to phone.
This USB Ethernet adapter has worked for me on several Google-branded Android devices: a Nexus 5, Nexus 5X, and Pixel C tablet. I've been told by a friend that it also works on the Wii U. It's an older device (note, doesn't support gigabit) and apparently pretty widely supported.
However, other Android device makers may have removed some drivers from the version of the OS that they ship on their devices, so there's no guarantee that it'll work with every phone.
I don't know about your phone but the Nexus 7 can be configured the way you want, view the following video.
Alternative Method :
Buy a phone which has a usb internet option in settings (like my htc 816g), and just connect your phone via usb to your computer and share computer's internet with the phone.
In both methods you are safe from radiations of wifi or mobile data.
Hope it helps.