As the title indicates,
- Can vice president/security advisor or secretary of state be chosen from the opposite party?
If so, has there ever been an example?
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As mentioned in comments, the Vice President was originally the runner-up in the Presidential Election; this practically guaranteed that (until this was changed) the Vice President would be of a different party than the President.
Now, because the Vice President is on the ticket alongside the President, they will almost certainly be of the same party, but nothing besides politics would prevent a joint ticket across parties. Bernie Sanders has not advanced to be the Democratic Party's nominee though he has run for it, but it's worth noting that he isn't actually a member of the Democratic Party, he's an independent. Were he to have won the nomination it seems likely he would have chosen a Democrat to be his Vice President.
As for other political appointees, Wikipedia has a helpful list. Walter Gresham, a Republican, served as Secretary of State for Grover Cleveland, a Democrat. Thomas Jefferson was also generally a political opponent of George Washington but served as Secretary of State prior to the development of the formal party system.
Many other appointments, even in recent times, have crossed party lines. Two different Republicans had time as Secretary of Defense while Barack Obama was president (Robert Gates stayed on from the same role in the Bush administration). Michael Flynn was very briefly National Security Advisor under Trump and reportedly considered for VP despite being registered as a Democrat, though it is difficult to associate his political leanings with the modern Democratic Party.
Appointments across part lines often involve people who are near the political center and/or have a long history of service in government itself rather than politics, such that their role is not as partisan. Others may be in their positions as part of a deal with an opposing party who controls the Senate confirmation process under the Appointments Clause, or in a more general demonstration of bipartisanship. And others, like Flynn, may have a party affiliation that does not particularly fit their politics, particularly if they have not served an elected office, or if parties realign.
The Constitution, as written, does not even allude to party politics. The 12th Amendment was written in reaction to the reality of political parties and ensures that in all likelihood, the President and Vice President will be members of the same ticket and thus likely the same party. But basically anything to do with political parties in America at the federal level is a matter of custom and practicality, not the law. The President has the power to make appointments, the Senate has to confirm them, and those are basically the only requirements for the position.
As for examples, there are plenty. But the most amusing example would be that Donald Trump's Special Advisor to the President was, for most of his term, a registered Democrat. Her name was Ivanka Trump.
Yes it is possible.
For the Vice President there is a specfic example of Andrew Johnson, a pro-union Democrat, who formed a "National Union" ticket with Republican Abraham Lincoln.
There is no special requirement that the Secretary of State is from the same party, although there is an expectation that Secretaries of State will promote the policies of the administration, and so most Presidents will choose someone who they can work with politically. Some Secretaries of State have been seen as moderates with links to both parties (Colin Powell for example has been touted as a Democrat VP nominee, before siding with the Republicans in the mid 90s), however I can't find an example of a President hiring a political opponent to this role. (Another answer mentions Democratic President Grover Cleveland's Secretary of State, Republican Walter Gresham, which was a good find, however by the time of his appointment Gresham was actively supporting Democrats, and had campaigned for Cleveland in the Election. So he was hardly a political opponent of Cleveland.)
And below the Secretary of State, it is possible for a President to appoint people without a strong political position, or to maintain their predecessor's appointment for the sake of continuity. An example would be Robert Gates, who served as Defense Secretary in both the Bush and Obama administration.