I just came around a corner and saw one Delamain-Taxi already upside-down on the street and also suddenly exploding. Then Delamain called and blast all missions of because I apparently used to much unneccessary violence. Is this a bug or part of the mission? Could I have done something to prevent this?

The car should not have been upside down and should not have exploded.

None of Delamain's cars are protected though, so they can be totalled like any other vehicle at which point you'll fail the mission.

Reloading often works; none of the gangoons or random events are inherently hostile to Delamain's cars, so if it was the victim of a random collision or shootout re-running will usually net a different outcome.