Do most commercial jets have a negative lifting body?
You are right, the upsweep at the tailcone allows the pilot to rotate the aircraft during take-off and landing without touching the ground with the rear fuselage. When such a tail strike  happens, the aircraft needs to be inspected and repaired, and incorrect repairs after a tail strike have already caused loss of life.

While most Boeing fuselages show some downslope of the upper tailcone contour, this effect is even more pronounced with all Airbus aircraft except the A380. There the upper contour of the tailcone is straight. Military transports with a rear ramp show even more upsweep to keep the heavy ramp short, and most need additional strakes to limit the area of separated flow there at low speed.

This upsweep adds a small downforce and, since it happens away from the c.g., a pitch-up moment. First, at the transition from the cylindrical section to the tailcone you find a strong suction peak at the high curvature region on the lower fuselage. Also, the lower side of the tailcone has a lower pressure than the upper side. Both create the downforce and the pitch-up moment. However, a three-dimensional body of a very low aspect ratio like the fuselage has a very low lift curve slope, so even the strong upsweep creates a small moment when compared to the contribution of the horizontal tail surface.

The local flow at the tail is dominated by the wing wake. To create lift, the wing has to deflect the flow downwards, and this produces a local flow direction at the tailcone which aggravates the situation created by the upsweep. The best indication for the local direction of flow is the rain diverter above the rear fuselage door. See below the rear door of the A320: To create the lowest possible drag, the rain strip is aligned with the local direction of flow during cruise. When flying at lower speed, the downwash is even stronger.



A320 rear fuselage (own work, based on this source)