Does the tailplane have a positive/negative camber? I've read that the tailplane produces negative lift, so does that mean it would function like an inverted wing (has negative camber)? The image below is the best illustration I could come up with.



Also, is negative lift and tail down force the same thing?
The tailplane should produce less lift than the main wing, for pitch stability. That is to say, its lift is negative relative to the wing lift. Its lift need not be wholly negative (pointing down), although it usually is during takeoff and landing.

But yes, as a first approximation, the direction of camber matches the direction of lift.

Also yes, negative lift (at the tail or anywhere else) can be called downforce (at the tail or anywhere else).

Here is Peter's explanation.