What is the electric field inside a cylinder?
The field everywhere inside the cylinder is zero.
Is the electric field outside the cylinder the same as if all the charge were on the axis?
Since the electric field must be constant everywhere on the tube wall, the electric field must be zero. Note that the result is the same as if the charge were concentrated in a line along the axis.
What is the field inside the inner conductor between the two conductors and outside the outer conductor?
The electric field is zero inside a conductor. Just outside a conductor, the electric field lines are perpendicular to its surface, ending or beginning on charges on the surface. Any excess charge resides entirely on the surface or surfaces of a conductor.