To correctly address this type of problem you need to set up an appropriate set of coordinates and evaluate all torques and forces in this coordinate system. The torque is defined about an axis, or relative to some axis via the relation cross(r, F), for each force. r is the vector from the axis to the point of contact of F. For "free" bodies one can separate the motion into two components, that of the COM which is governed by the net force, and the motion about the COM which for a rigid body is pure rotation and governed by the torques. When you constrain a rigid body to be fixed at a point (spherical hinge) or along an axis (like a door) you could still describe the motion about the COM but that becomes counter intuitive. A better approach is to calculate everything relative to the fixed axis of rotation (in this case defined by the hinges). In this description there is one and only one degree of freedom needed to describe what is happening (you really have 3 degrees of rotation but the hinge constrains two of them).
The force you apply produces a torque about the axis defined by the hinge(s). There is a reaction force at the hinge (there has to be for the constraint to work). Consider a figure of the door with a hinge modeled as a cylindrical post passing through a cylindrical cuff (hole made through the door) and consider an infinitesimal gap between the post and the surface of the cuff. The hinge force is a contact force. As such there are only two possible contributions to that force (in our ideal model where the door and hinge are "rigid"). The first is a Normal force due to the contact of the two surfaces. This will point along the radial direction along a line through the center of the hinge. The second is traction between the post and the cuff, i.e. a grip tangent to the surfaces which is due to friction. The first, being normal, will never produce a torque as cross(r, F) = 0 for that force. The second will produce a torque that resists the force you are exerting to open the door. If this happens you should apply oil or WD40 to the hinge. We can assume a friction-less surface between the post and the cuff and that force goes away. In a real life situation this force should be very small or can be made arbitrarily small. Hence relative to the fixed axis of rotation the hinge produces no torque. You can also take the limit of a very small radius for the hinge and arrive at the result that the torques due to the hinge forces about the rotation axis are approximately zero.
One key to understanding the different approaches to describing the situation is that you are free to evaluate the torques and the motion in any coordinates you want.
For free objects the COM frame is ideal for describing rotation, for fixed bodies the fixed axis is ideal.