Re: Ten Commandments - the right to display in public places
The rules are much more nuanced than your question suggests, but they stem from the Estagblishment Clause of the First Amendment to the United States Constitution. That clause says "Congress shall make no law respecting an establishment of religion . . . ." That clause has been interpreted to mean that the government cannot do anything that endorses any religion over others, or that endorses religion over non-religion.
Installation of one religion's symbols in a government building suggests that the government backs that religion, especially when symbols of other faiths are not present and when there is no other purpose for the display. This concern is especially important in courthouses, where litigants, criminal defendants, jurors and witnesses are forced to appear and thus have no way to prevent being exposed to this endorsement.
Since the majority of Americans are Christian, most of the symbols which have been placed in government buildings have been either Christian (e.g., a crucifix) or Judeo-Christian (e.g., the Ten Commandments) in nature. Since many Christians have no problem with such displays many of them don't understand why they are controversial, but most surely would see the problem if government buildings were instead displaying only the symbols of Hinduism, Buddhism, Islam, Shinto or any other non-western faith.
The reason you didn't have a chance to vote on it is that it was written and approved by Congress in 1789 and then ratified by the states in 1791. By definition, a constitutional amendment cannot be unconstitutional.
Of course, since the clause is only ten words long, different people have interpreted it in different ways. Supreme Court decisions have formed the dominant interpretation over the centuries, and many of these decision were influenced by a letter Thomas Jefferson wrote to members of Congress in 1802 in which he said the purpose of the clause was "building a wall of separation between church and state."
The current interpretation is probably different from what the founding fathers had in mind, but then our society is different from theirs; almost every American citizen was Christian 200 years ago and the framers were mostly concerned about the government promoting one version of Christianity over others. Whether they would approve of the changing interpretation over time in light of the country's changing demographics is hard to say.