Background

Impeachment in the United States is mostly handled by the legislative branch alone. Hence the Courts have not typically applied restrictions or said much about impeachment other than Nixon v. United States:

Primary Holding

The political question doctrine is triggered when the text of the Constitution has shown that an issue lies outside the scope of the courts, or there is no judicial standard for resolving the issue.

This is the only Supreme Court case that I know of which directly applies U.S. Const. Art. I, Section 3, Clause 6 (i.e. the Trial of Impeachment) in any context. I would like to know from a legal perspective any other limitations and or interpretations that the Supreme Court has given to U.S. Const. Art. I, Section 3, Clause 6. Hence my question

Question

What Supreme Court cases, other than Nixon v. United States, have directly applied or interpreted how the Trial of Impeachment works on the federal level?

Actually this is the only SCOTUS ruling on Impeachment because of what it legally means with respect to SCOTUS and impeachment. Namely, Impeachment is a congressional power and not a judicial one that has no punishment beyond the removal from office upon conviction and that therefor it is not a matter that is Judicial. That is a fancy way to say that SCOTUS or any lower court may not hear appeals rising from the outcome of Impeachment. Because of this, it's unlikely to hear any SCOTUS case as to the matter of impeachment and the only outstanding question they could likely hear is "Who can hear an Impeachment Trial for the Vice President?" but thus far none of the 19 impeachment trials have heard involved a vice president. Nixon v. United States basically held that in matters of impeachment it is the court's position that it is likely to respond to this hypothetical with the legal ruling of "not my monkeys, not my circus" (note: not legal jargon).

With that said, this makes the U.S. senate the highest court in the land when it comes to precedents set by impeachment cases. The very first person ever impeached was a U.S. Senator, who was expelled by the Senate the same day the House voted on Articles of Impeachment. The decision not to have the trial did establish some notable precedents: first, Impeachment and conviction are all about removing an officer from office and if at anytime before conviction the officer is removed by other means, the process is stopped as it is moot. Second, and more important, is that members of Congress are not "impeachable" officers as both houses have methods that allow them to expel members by vote. This means that the only people who can be impeached are executive officers (the person who is currently president, vice president, and cabinet secretaries, any other office that is appointed by the president and confirmed by the senate, and federal judges) and that a conviction of impeachment does not bar one from holding another federal office unless the senate enacts an additional punishment after the conviction barring that person from federal office. They cannot bar one from state office. Finally, House and Senate Rules have instructions for procedures in their part in impeachment. The commonality between both is that impeachment takes precidents over all regular buisness so once articles are put to the floor, there speaker must hold the vote with in a certain window of time (three days, I think) and the Senate must hold the trial as soon as possible, though will honor reasonable delays as per the impeached officer's right to delay.

As a final note, and because it wasn't clear in the question Nixon v. United States is often confused with another case (United States v. Nixon). The latter one did indirectly relate to impeachment in that it was related to President Nixon handing over evidence to the comittee investigating possible impeachment articles against Nixon, and because of that ruling, Nixon handed over the evidence and later resigned to avoid the Impeachment.

The former one not only didn't involve anything related Watergate, it also wasn't related to President Nixon at all, but a (former) Federal Judge Walter Nixon, and specifically was heard because Judge Nixon happened to be the first Judge impeached by a Senate commitee as the jury rather than the full Senate, following a rule change that only held full senate as jury for the President and Vice President and a committee for anyone else.

The final rule is that only the President is constitutionally mandated to have the Chief Justice of the Supreme Court act as the judge in an Impeachment trial, while the presiding officer for all other cases is either the Vice President (acting in his role as President of the Senate) or the President pro Tempore of the Senate (acting in his role of "Guy Keeping the Vice President's Seat in the Senate Rotunda Warm" (not technical Jargon)). Which is why the question of "who is the judge in impeachment of the Vice President?" a noodle baking question, and the answer differs from the Chief Justice to the Senate pr Tempore to the rarely serious argument of the Vice President himself. The best answer is that "It has never come up."