Do any airplane designs exist that don't involve a flight surface that provides downforce?
There are multiple configurations which are possible with tail or canard, which based on their locations and whether they produce lift or down-force, results in a stable or unstable aircraft (taking the aircraft center of gravity into account). The figures below show some of the possible configurations.



Source: f-16.net

In the most general case, there is no need for even a canard or a horizontal tail as long as a (down or up) force is produced in some way. A good example is the Boeing Bird of Prey, where the chines produced (vortex) lift and thus a stabilizing force. The forces acting on the longitudinal axis of the aircraft look something like this, leading to a statically stable aircraft without any lifting surface except the wing.



Profile from bj-o23.deviantart.com, others own work.