If a president is impeached, can they be further charged after the impeachment, and if so, are they charged as a citizen or as president

The President can be charged after impeachment. Impeachment is just removal from office

Thanks- but is he charged as president for the trial, or as an ordinary citizen

Impeachment is NOT removal from office. Impeachment is basically indicting an official on charges. A trial for the offenses comes after Impeachment, and the result of that trial can include removal from office.

This is why Clinton was impeached but still remained in office, as he was Acquitted on trial.

I have no idea if double jeopardy applies. My guess would be it doesn't (as its not actually a federal court but a trial by the upper house of legislature) but I don't know for sure.

What if Bush shot Cheney to death live on tv, would he be arrested by the police? Or would there have to be an impeachment proceeding?

I think Cheney would be charged with a crime and arrested by the Secret Service. He would probably be impeached as Vice President, as he would never be sworn in as president, and the Speaker of the House would be sworn in as president.