Impeachment is not removal from office. Trump was already impeached last year and still ran for re election. So impeachment won’t matter unless Senate votes to remove.
Right, the House votes to impeach. Impeachment is kind of like an indictment. There’s a legal saying that you can “indict a ham sandwich”, which means that it’s fairly easy to indict someone even with a weak case.
The Senate holds a trial and can convict with 2/3. If the Senate votes to convict, they can remove from office. The Senate can also vote to prohibit the person from holding future public office.
Alcee Hastings is an example of a federal judge who was impeached and removed from office, but the Senate did not disqualify Hastings from serving in a future public office. Now he’s a Congressman.