Yes, but a weaker one than in lengthwise direction.
You can observe the local pressure by comparing the distance between two stream lines. When they move closer together, speed goes up and static pressure goes down. As you move away from the airfoil, the distance between stream lines recovers to its value at infinity.
Stream lines around airfoil (picture source).
This effect depends on the Mach number of the flow. At very low speed, the recovery is rather quick because speeding up contracts the airflow so it is easy to make way for the airfoil. At higher (but still subsonic) speed, speeding up is coupled with a drop in density, so the contraction is less pronounced and it takes more height for the disturbance to die down. At supersonic speed, the pressure change is sudden (in a shock or an expansion fan), and the pressure rise of a shock reaches into infinity.
Yes and no. There must be a gradient between the low pressure at the surface and the atmospheric pressure in the farfield. However, for boundary layer analysis (Blasius solution), one of the key assumptions is that there is no pressure gradient in the wall normal direction.