Why do you use an additional object for the sensor “ground_chesk” and type - Collision? Is not it easier to create a “Ray” sensor on the player itself
Everyone has its own way, but rays cast from origin, so standing on a ledge while the origin is not on it (so 1/3 of the player stands on a edge) the player will stop reacting because the ray wont get into positive state.
another reason, the movement feels heavier this way, while climbing and walking angles.
physics bunny hop problem is solved with the sensor as well look at here:
and here i solved the problem
some like rays, in this case you need rays to be cast from 5 points, center and 4 corners of the player, a sensor well you put it there and it works, not much work and look at tha last movie you see how it reacts to slopes as well, no flying etc just relaxed walking up and down, no bouncing either.
And when running through an external player, the player and camera are shifted to the right - 2.78с
the blend you uploaded has nothing wrong, works as it should here.
but if you got shifting problem i bet you moved the character, the camera is on slow parent, if you move it all cam settings will be off. in order to move em you select the camera empty go to objects tab in properties panel, scroll down to relation extras and turn of slow parent.