Yes, anytime I activate the 3rd device the other two get disconnected. I wonder if there's a limit of 2.
I've never owned more than 3 Rokus. In fact they're still the original streaming sticks; I've never upgraded them because they work just fine. I don't have any additional Cox usernames nor have I ever given out account information to anyone. I take care of everything so the household just has to fire up the Roku and start streaming. I don't own a smart phone (nor does anyone in the house) so I can't test your phone app theory. And, yes, having an authorized devices list in my online Cox account settings would be grand.
We're a household of seniors and we don't go out much except for doctor appointments and grocery shopping so TV is our major entertainment.