What, if any, are the limits on an impeached and not convicted president?
There is indeed no change in the president's official functions/capacity until he is convicted by the Senate and removed from office (which is done in one and the same vote).
A president can continue governing even after he or she has been impeached by the House of Representatives. After then-President Bill Clinton was impeached on Dec. 19, 1998, he remained president for another year, during which time he was acquitted in a Senate impeachment trial. While Clinton continued governing, and the impeachment had no legal or official impact, his legacy is marred by the proceeding.
The Senate can also bar him from running again for presidency, but that's a separate vote, which is optionally held only after conviction/removal.
However, the impeachment proceedings do give justification to the House to subpoena a lot of people close to the president. That can be at least an inconvenience (for him) if not outright trouble if more embarrassing material comes to light.
Already we have news that:
US Secretary of State Mike Pompeo has accused Democrats of bullying his staff as a part of an impeachment inquiry into President Donald Trump.
He said in a tweet that requests for five officials to appear before a committee were "not feasible".
Nancy Pelosi hands out subpoenas 'like cookies': Trump
Also, this is not a 100% obvious question. In some parts of the world, a president can be suspended while under impeachment/investigation. I remember reading this about Romania's impeachment of their president in 2012, where their parliament suspended their president. Over there they need[ed] a referendum to actually remove him from office... and their president at the time survived the latter.
This is not quite on point, but is probably of interest.
Richard Nixon was not impeached because he resigned before it happened, but he certainly knew he was about to be impeached, and there was little chance that the Senate would acquit him.
I am not aware of any official limits that applied to him during this period, but the Secretary of Defense did discreetly order military commanders not to carry out any nuclear launch orders until they had checked with either him or the Secretary of State. Evidently he feared that Nixon, who was depressed and drinking a lot, might decide to commit the ultimate murder-suicide by starting World War III.
To understand what happens it probably helps to understand the meaning of the word "impeachment" outside of the legal system.
According to MW, there are 2 meanings of the verb "impeach:
- to charge with a crime or misdemeanor
- to cast doubt on
It is the 2nd meaning that is the plain-English sense of the word. And the Constitution used it in the plain-English sense. The word is still used in this way in fact-finding of any kind. "Unimpeachable" is used to refer to evidence which has no reason to be doubted. To "impeach" a testimony is to show that there is a reason to doubt it.
How does this effect the answer to the question? Well, the House of Representatives "impeaches" and the Senate "convicts." What this really means is that the House shows reasons for why a politician (who is subject to impeachment) should be put on trial. The House creates an accusation. If the Senate convicts, the politician is removed.
This is not a process of removal from office. Nor is it an impeachment. Removal from office is something that happens at a point in time -- not over a period of time. Impeachment is what the process is colloquially called. It is called that only because the process is so rare that there is no convenient language to colloquially name all the parts of the process.
This should make the answer to the question in the title obvious. What should happen to someone accused but found, as a result of a trial, not guilty? Well, nothing.