Why do airfoil's trailing edges point downward?
You are right, the trailing edge does not need to point down. Take symmetric airfoils - here the trailing edge runs parallel to the airfoil chord. Or take reflex airfoils (like the HQ 34 of the SB-13 tailless glider): Here the trailing edge is indeed pointing upwards, and still this aircraft flies.

But to create lift with as little drag as possible, it helps that the trailing edge points slightly down. Why? Because then it will be least in the way of the desired local flow. Lift is created by accelerating air downwards. The wing deflects the air flowing over it, and the trailing edge should reflect this deflection angle. 

But - as always - too much is not good: The Eppler 417 airfoil of the SB-7 glider was an extreme layout with too much rear camber. It is said that pilots could still wipe off raindrops near the trailing edge after landing when they had crossed a shower an hour earlier. This means that airflow separated well ahead of the trailing edge on the upper surface, and the raindrops would not be blown away in the separated flow. Flow separation increases drag, and this effect shows that the shape of the Eppler 417 had too much downward camber at the trailing edge. 

If the plane is large and heavy, its airfoil's rear camber can be higher - an extreme case is shown below. This is an early supercritical airfoil designed by McDonnell-Douglas, and the highly cambered rear part allows it to integrate very effective Fowler flaps. Flaps help because they allow to change the direction in which the trailing edge is pointing: Low lift coefficients require no flap deflection, or even negative flap angles in gliders, and the higher the lift requirement grows, the more the flaps will be extended, pointing more and more downwards.



The same is true for control surfaces: Depending on the desired lift change, their trailing edge will point up- or downwards. See below for an example where I plotted the pressure distribution for three flap deflections in one graph. Upper and lower surface pressure are shown by color-coded lines, and lower lines belong to the lower surface. Dashed lines show the inviscid pressure, and solid lines the pressure distribution with friction effects added. The wider two lines of the same color are apart, the more lift is created. Note the contour plot below, which follows the color scheme of the pressure plots.



If there are slots to re-energize the flow, extreme trailing edge angles are possible and help to create lots of lift at low speed, which helps airliners to get into small airfields. See below the triple-slotted flaps of the Boeing 727, which was designed for regional traffic from and to small airports.