The answers to your questions follow:
In the first sentence, the correct answer is “What right does the government have . . . ” since government is a singular noun and requires a singular verb. In the second sentence, the correct answer is “The technical data submitted by Buick do not have any markings.” The reason is that data is a plural noun and requires a plural verb. The plural verb in this case is ‘do’. It sounds strange to the ear because most people don’t know that data is plural for datum.