Is the distance on a B737 engine really that little comparing to other Aircrafts (for example A320)?
The actual distance depends on the engine installed and the weight and balance of the airplane at the time. But there isn't a huge difference between the two planes. Comparing the figures, the current A320 is about 10 cm higher than the current 737 NG. The A320neo and the 737 MAX will be pretty close, with the MAX potentially getting 5 cm lower.
737 (MAX): 43-58 cm (1.42-1.92 ft)
737 (NG): 46-64 cm (1.5-2.08 ft)
737 (CL): 46-53 cm (1.5-1.75 ft)
737 (100/200): 51-58 cm (1.67-1.92 ft)
A320neo: 46-56 cm (1.84 ft)
A320: 57-85 cm (2.2-2.79 ft)
Those reports also show exhaust velocity profiles at takeoff thrust, but differences in the diagrams make it hard to compare. The IAE engines look to have much lower velocities than the CFM engines on either plane.
Is this a design fault? Also have a look at the "Jet-blast pressure study" page 13-14.
It depends how you look at it. In the linked report, there are several recommendations for both blast pads and the operator's procedures, but nothing regarding the design of the aircraft. So rather than telling the designers to mount their engines higher above the ground, they are making sure that facilities and procedures are designed to prevent things like this from happening.
As far as being a "fault" unique to the 737, the study mentions:
...an accident very similar to that of G-DOCT, which involved an Airbus A320...
So it's not the only airplane that can cause problems. And the newest versions of both types will have similar and smaller distances, they probably decided that there is no facilities-related reason important enough to choose differently.