If, after election and before inauguration, a US president elect is indicted under federal law, what would happen to the office?
@cpast's answer is correct as far as it goes. An indictment would not prevent a a president-elect from assuming office. In that situation, Congress could impeach the new president or refuse to certify the election results (which would provoke a constitutional crisis).
It seems a bit unlikely though. An indictment (or even an investigation) before the election would be more effective at keeping that person from being elected. If done soon enough, the candidate's party could replace her or him on the ballot. If done just before the election, this could easily cause that candidate to lose. So this would generally be done earlier in the process.
An indictment is just a formal accusation that a person committed a crime. It does not mean that it was already proved that the person is guilty. In a society which respects rule of law (like the United States) a person is considered innocent until convicted by a court of law. So the indictment alone would not have any consequences whatsoever for the status of the president, except for being forced to follow instructions by law enforcement. In theory it might even come to the weird situation that the president is forced to lead the country from their jail cell while waiting for the court case to finish.
Curiously even a conviction of a major crime does not yet mean that the president needs to step down. However, being convicted of "treason, bribery, or other high crimes and misdemeanors" is reason to start the formal impeachment procedure which eventually results in removing the president from office if congress and senate agree to do so.
This depends. If the President-elect is arrested but not yet convicted, he or she is still capable of entering the White House. However, many of his/her privileges would (could) be restricted, such as his/her security clearance (see below). Knowing that an arrest might hinder his/her ability to do the job, he/she has two effective choices: (1) move forward with the swearing-in ceremony, allow the judicial process to continue, and if found guilty, almost certainly be impeached while simultaneously pardoning him/herself; (2) move forward with the swearing-in ceremony and resign shortly thereafter, giving the White House to his/her Vice President.
Here is where it gets tricky:
One thing that would probably not happen is a step-down after election but before the inauguration because that would place the decision of the Presidency in the hands of the House of Representatives. In other words, it is treated as if neither party won the electoral college. The President-Elect cannot turn over the White House to his/her VP because he/she has not yet been inaugurated, thus that power to resign is not yet vested (i.e., you cannot resign from that which you do not yet have).
Another curve ball...
If the President-Elect is arrested for a federal statute outside of the scope of judicial process (e.g., mishandling classified documents, 18 U.S. Code § 798) that could eliminate a trial process [believe it or not, this happens]. If an executive statute is broken (CFR), the President is obligated through the presiding cabinet official (Secretary of State... Defense... Attorney General, etc) to deny the President-Elect executive privileges typically consistent with the Office of the Presidency (this is called debarment; not to be confused with disbarment, a procedure reserved for lawyers). There is no right to judicial process - as an American - if you are arrested and determined to have broken an executive statute. It is the only process in the United States within which someone accused is presumed guilty and must prove innocence during a review run by technocrats. If the President fails to act, the Senate could demand the Sergeant of Arms remove the President-Elect's privileges by a straight up-and-down vote. At that point, the Sergeant of Arms literally bars the President-Elect from entering the White House until he/she can be removed by Congress or he/she resigns.
The latter would be a bad day for the United States...
A president cannot be impeached for crimes that occurred before they became president. If the indictment comes before the election, there are many scenarios that could take place.
1. They could drop out voluntarily
2. Their party could replace them with whomever they all agree on.
3. They could be elected and the electoral college could reject the outcome.
4. They could be elected and after they are sworn in they could pardon them self.
The current president would nullify the election results and call for a special election and would vacate the office of the presidency after the new election results are formed. How long that might be who knows. We will see claims of dictatorship for sure. We might be seeing a historical event happen here with the current election and its scandals