For example, could Bill Clinton have been charged with perjury in the standard judicial system, even while sitting President? Or is impeachment and removal a prerequisite for judicial proceedings?
Governor Rod Blagojevich was charged and released on bail prior to being impeached and things going to trial. Holding office does not grant immunity, nor does impeachment automatically imply removal from office, and neither are a prerequisite for criminal proceedings to be undertaken.
Governor Evan Meecham of Arizona, for example, was not even impeached until after his conviction.
Here's a short set of details on several gubernatorial impeachments
Article I, Section 6, Clause 1 of the United States Constitution says
"The Senators and Representatives [...] shall in all Cases, except Treason, Felony and Breach of the Peace, be privileged from Arrest during their Attendance at the Session of their respective Houses, and in going to and returning from the same; and for any Speech or Debate in either House, they shall not be questioned in any other Place.
This prevents them from most cases of criminal liability for things they do during debate and prevents them from being arrested during debate or while traveling there, but doesn't offer them any protection outside.
No such rights apply to the president. In fact, Article II section 4 of the US constitution explicitly states that: "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.". If an impeachment would be a prerequisite for conviction, this section would be redundant. If "perjury" is a "high crime or misdemeanor" warranting an automatic impeachment would be for a court to decide. There is no precedent for this, because no president or vice-president was ever impeached for criminal activity (Nixon resigned voluntarily).
This is different from many other democracies where it is common that parliament members have immunity from criminal prosecution unless the parliament revokes that immunity with a majority vote (which they usually do when there is reasonable ground for suspicion). The reason for parliamentary immunity is that the executives in some "less democratic democracies" like to abuse their authority over the law enforcement system to arrest parliament members who make uncomfortable voting decisions.
A US official can be charged and tried in court without going through the impeachment process at all.
- Consider Scooter Libby. He was a US official. He was indicted on multiple counts. He resigned. He was convicted. His prison sentence was commuted but his conviction remains and he still has to pay several hundred thousand dollars in fines.
- Consider Richard Nixon. He was a US official. He was pre-emptively pardoned. He likely would have been charged and convicted for multiple felonies without going through the impeachment process.