To what extent did the construction of the Berlin Wall lower the temperature of East-West rivalries?
The construction of the Wall did a fair amount to lower the temperature (assuming that you mean that it reduced tensions). As John F. Kennedy once said, it was better to have the wall than to have a war.
Before the wall went up, there were literally millions of East Germans escaping to the West. The leader of East Germany had been pushing the Soviet Union to do something about this. One of his suggestions was to simply take West Berlin by force. As long as there was no wall and escape was so easy, this sort of a strike on the part of the Soviets was a real possibility. It was also possible (and this is said to be a reason the Soviets allowed the wall) that the East Germans would force the Soviets to do something by attacking on their own.
Because the wall took away the danger of this sort of attack, it lowered the degree of tensions between the East and West.