The constitution give the senate power to remove him from office and possibly bar him from serving again. That's ALL the senate can do. However, the constitution leaves the convicted person open for civil charges and trial, not by congress but by the civil courts.
It's been held that person convicted after impeachment can't appeal. However, that's not really been tested. If congress violates the constitution, the SCOTUS is supposed to rule their actions invalid and in the past they've done so many times. Why wouldn't that hold here?
The constitution specifies the procedure for impeachment. Chapter and verse. The constitution only in general and broad terms specifies what may constitute cause for impeachment. Thus, the House of Representatives would have to be incompetent and not have access to a competent attorney in order not follow a pretty simple procedure. The general and very broad definition of impeachable offense makes it next to impossible to wrongly charge the impeached party. So, about the only way for a trial after impeachment to be unconstitutional is for the person presiding over the the trial to engage in or permit blatant unconstitutional action. In the instant trial, Trump was impeached prior to his term expiring for actions committed prior to his leaving office. To not be able to hold the trial deprives the nation of it's ability to hold the president accountable for his ENTIRE term, and deprives the president of his right to answer such charges.(to confront his accuser). Both are long held principles in our legal system.
Undoubtedly the rationale for the Chief Justice presiding at the trial is to assure a trial procedure that passes constitutional muster.
Since an impeachment is a political action by design and not the creation of a law which applies equally to all citizens of the nation, the Supreme Court nor any other court has any say in the trial or it's outcome.
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