I am curious, is there anything preventing the President and Vice President from switching positions?
For example, could the President resign, and then the newly elevated VP appoint him to his old position?
Could this be done to avoid the effects of an impeachment?
There is no law preventing this scenario from happening as you describe so it could happen in theory. The part that would make this practically impossible is that the VP appointment would require both houses to consent due to the 25th ammendments. It's very unlikely they would ever consent to an appointment of a president who resigned, especially one that was impeached.
In a non-scandal situation, this might be possible (although weird), but this wouldn't be an effective way to avoid impeachment.
First of all, resigning doesn't necessarily stop the impeachment process. The Senate voted in 1876 to proceed with the impeachment trial of Secretary of War William Belknap, despite the fact that he had already resigned (he was ultimately acquitted).
Secondly, an impeachment proceeding can (optionally) disqualify someone from holding any future public office. If the Senate voted (subsequent to removal) to do that, they wouldn't even be eligible to be appointed to VP.
Thirdly, according to the 25th Amendment, a Vice Presidential appointment is subject to confirmation by the Sentate both houses of Congress (thanks user102008). So unless there was a full turnover in Congress, the prospect of them confirming someone they just removed (or were starting the process to remove) seems vanishingly unlikely.
Finally, Congress could still impeach a newly appointed VP, even for things they did before entering that office. See page 15 of this Congressional report on Impeachment (PDF):
Judge Archbald was convicted on four articles alleging misconduct in his then current positions as a circuit judge and Commerce Court judge, and on a fifth article that alleged misuse of his office both in his then current positions and in his previous position as U.S. District Judge.
So you could still impeach that new VP for his prior conduct as President, or even his conduct after he resigned from office.
And I would imagine that if he had already been impeached as president, it might be possible to repeat that process very quickly, based on all the prior investigation and evidence (and then invoke the clause to disqualify them from holding any future office).
This could not happen to avoid impeachment (The Democratic Speaker of the House said he wanted the VP position filled before continuing for Impeachment so that it didn't come off as a coup to Republicans following Spiro Agnew's resignation from the position).
That said, the VP can become Acting-President for a time where the POTUS is not able to perform his duties but will return at a later time (George H.W. Bush after the attempted assassination of Ronald Regan as well as Dick Cheney during surgeries that George W. Bush underwent.). Upon return to health, POTUS resumed duties and the Acting-President was demoted to VP.
