Why are two pairs of a wheatstone bridge of opposite direction?
I wonder why are, e.g. when using a wheatstone bridge in combination
with strain gauges, two of these SGs do have a positive k-factor and
the other two a negative k-factor.

A typical example is on a cantilever beam where a strain gauge on the upper surface is matched by a strain gauge on the lower surface. When the beam is bent upwards the upper gauge is compressed and lowers its resistance whilst on the underside, the gauge is stretched and increases its electrical resistance: -


Why is the resistance change not zero then in the end? That is, when
R1 is increasing because the SG is lengthening, and R2 is decreasing,
because the SG is shortening, then there should be no change in
resistance at all

Absolutely correct but, that is of little importance given that we are largely uninterested in the top to bottom net resistance of the bridge; what we are interested in are the voltage changes at the midpoints of each limb due to (a) resistance lowering on the compressed gauge and (b) resistance increasing on the stretched gauge.
I think I see where your diagrams have confused you - basically, the RH diagram is wrong as depicted below: -

The upper right should be in tension and the lower right should be in compression to make it work properly.