If a president is impeached, convicted and removed, what comes next?
Is there a general election? Does it go to the former-president's VP? And what happens to his or her appointments and administration?
There won't be a general election. The succession will continue to follow the line of succession as established in the Twenty-fifth Amendment to the United States Constitution.
Regarding the cabinet appointments, the new President can choose to replace or retain the officials in the previous cabinet. This happened before when Nixon resigned and his successor Ford replaced most of Nixon's cabinet members.
The reason that the VP succeeds the President is that only the President is impeached and convicted. The impeachment does not include the Vice-President or his cabinet officials.
The Twenty-fifth Amendment states that "In case of the removal of the President", the Vice-President would succeed him/her.
In Case of the Removal of the President from Office, or of his Death, Resignation, or Inability to discharge the Powers and Duties of the said Office, the Same shall devolve on the Vice President, and the Congress may by Law provide for the Case of Removal, Death, Resignation or Inability, both of the President and Vice President, declaring what Officer shall then act as President, and such Officer shall act accordingly, until the Disability be removed, or a President shall be elected.
In case of the removal of the President from office or of his death or resignation, the Vice President shall become President.
Note: The Twenty-fifth Amendment supersedes the wording of Article II, Section 1, Clause 6 of the Constitution as it does not explicitly state whether the Vice-President becomes an Acting President or would succeed the President.
The President, Vice President and all civil Officers of the United States, shall be removed from Office on Impeachment for, and Conviction of, Treason, Bribery, or other high Crimes and Misdemeanors.
The US Constitution says differently. All civil officers including the Vice President all go. The next in line would be the Speaker of the house, Paul Ryan (ugh) Why are you saying otherwise?