The wall was built by the GDR with prior consultation with the Soviets. There was no joint effort to ease tension and the move was condemned by the West in general and actually raised tensions.

I have never considered the question how much foreknowledge western governments had, but this German newspaper article claims that the BND, i.e. German intelligence, became aware of the plans three days before the event. While the idea to seal off East Germany was floated for quite some time the actual decision was made at the beginning of August 1961 in Moscow when Walter Ulbricht met Khrushchev to discuss the exodus of East Germans.

Officials in West Berlin protested pretty much every time whenever someone was shot at the wall, whether it was someone fleeing from the East or someone from the West accidentally or purposefully entering the East. As Berlin had a special status within Germany (nominally the city was governed by all Allies including the Soviets) any diplomatic protest had to come through the Western allies. Funerals for people shot at the wall, as long as they were buried in West Berlin, were regularly attended by high ranking city officials and used to condemn the very existence of the wall.

Cases where people from West Berlin were shot usually involved someone who was drunk or disoriented and accidentally crossed into East German territory. Other cases very early on involved people trying to help East Germans flee. They sometimes even exchanged fire with the border guards, something the West Berlin police did too. Other than that it was generally easy for people from the West to cross legally. The majority of people who died tried to flee from East Germany.