Coax. Connecting the outside cable coming up out of the ground to my inside wire at the box outside my house.
I have "certified 6A" ethernet cable.
Is there any reason to connect my 2 Sony's to the ethernet?
Any sort of weatherproof coupler that's of high quality should be fine. What's currently there? As @mk378 noted, make sure the cable is well shielded RG6.
On the Sony question, if these TV's don't have WiFi and subsequently aren't on the internet, yes, connecting them will allow them to get updates and stuff.