Why do wall-to-wall airfoils in wind tunnels produce no (or infinitesimal) downwash?
Who says that the flows only interact at the trailing edge?

The flow on each side of a wing creates (and is the creation of) a pressure field which influences what goes on on the opposite side in subsonic flow. Only in supersonic flow is that kind of interaction impossible because small pressure changes move only at the speed of sound.

First to the topic of flow turning. Half of that has already happened at the quarter point of the wing chord, so it is spread out over the whole wing. At the trailing edge the local flow direction is parallel to the trailing edge inclination if flow is not separated already. Past the trailing edge the flow readjusts to the downwash angle which is normally largest behind the center wing. In the case of an elliptic lift distribution, the downwash angle is constant over span, but it is almost never strongest near the wingtips. The experts at NASA certainly know that, too, but I wonder how often they are consulted by the marketing people who write those web pages.

Next to the tip vortex: It does not cause induced drag and is only a side effect of downwash. Please read this answer and try to follow all the links. There is not a single vortex but a vortex sheet leaving the wing, and to better understand what is happening it is best to forget all that.

Now to the conditions in a wind tunnel. The flow there is practically like one in ground effect, so no full downwash like in free flow is produced. In order to arrive at realistic results, wind tunnels use correction factors in order to make the results of their measurements close to those in free flow. If there is a gap between the trailing edge and the wind tunnel wall (and there certainly is), there is also some downwash, but less than in free flow. As explained in this answer, the part of the wake which flows down past the wing is smaller than the wingspan, so a crippled version of the wake behind a wing will also form in a wind tunnel, with air near the side walls moving up and air in the center moving down. Normally, this is all equalled out when the air flows through the next grating which is installed to remove turbulence or help the tunnel flow around the next corner.


  the explanations above describe downwash as the main (almost only) reason for induced drag and tip vortices as just a result of downwash.


Yes, tip vortices are indeed a result of downwash. You link already to two good explanations for that, so I need to go beyond those answers. The two vortex cores resulting from the rollup of the vortex sheet trailing behind the wing are not really the tip vortices you might see in humid air, like in the picture of an A340 below (picture source).



The condensation at the wingtip is from the high suction caused by flow locally negotiating the small radius of the winglet tip. This is the tip vortex. The much stronger vortex pair forming behind the wing is what the NASA page refers to, but that only happens when the air is already long past the wing. That is the wake vortex. In the cores of the wake vortex pair the pressure is lowest because the radius of the helical motion is smallest - vortex strength is the same for every point in a plane perpendicular to the direction of motion of the wing. Therefore, the NASA page is at least misleading, if not outright wrong.

In the wind tunnel the same happens (minus the tip vortex), but boxed in by the wind tunnel walls. They slow the wake vortex by wall friction and stop the wake from continuing downwards, just like the ground does in ground effect. But there is still a pair of vortices close to the bottom wind tunnel wall (or the top, if the wing section is mounted upside down).