The key point is that upon leaving the trap, the horizontal pipe cannot exceed 1/4" per foot of pitch before the vertical vent take-off.
NOT true, but the slope cannot be such that the vent connection occurs below the weir fo the trap. There is NO distance that will convert an "S" trap to a "P" trap without adding a vent. IF there is no vent then it is NOT a "P" trap regardless of what YOU call it. You also CANNOT connect it to the "drain line" with a sanitary tee since it is a horizontal line.