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.