I can't think of anything that does specifically 360 degrees but they may exist...it's basically a matter of the way the gearing is arranged. But finding them is difficult because as you've seen people use "360 degree" to mean continuous rotation (which isn't really a servo at all, just a geared motor).
Yep sure , I’ve some that do . You need the range of PWM signals to drive it this far .
There maybe an internal stop , ( which you can modify) but travel is ultimately limited by the travel of the feedback pot.