I heard Congressional Democrats are filing an emoluments lawsuit against Trump.
I don't get this. Wasn't prosecution of "high crimes and misdemeanors" (aka impeachment) in Congress's own job description and jurisdiction, not in the judicial branch's?
How can Congress ask the judicial branch to remedy a Constitutional issue regarding the President?
Anybody can ask the courts to remedy a legal or Constitutional issue regarding the president, if they feel it has harmed them. People sue the president and other executive branch officials all the time. Members of Congress have this right just like everybody else.
This lawsuit isn't alleging "high crimes and misdemeanors" and it's not seeking the President's removal from office. As you say, the judicial branch doesn't have the power to do that. Instead, the lawsuit is asking the court to find that the President's conduct does violate the Constitution, and to issue an injunction ordering him to stop doing it.
The reason why this case is being brought by members of Congress has to do with the question of standing. Normally, you can't sue over something just because you don't like it; it has to actually have harmed you in some concrete way. (Or rather, you can sue, but if the court finds that you don't have standing, it will dismiss your case.) In this case, the plaintiffs argue that their positions as members of Congress does give them standing. I am not sure I completely understand their argument; but the court will consider it and dismiss the case if they don't.
At a very simple level, when someone does something illegal, you might want to (1) force them to stop doing it, or (2) punish them. These are two separate things. For the President in particular, anyone has the right to sue in court to try to achieve (1). Congress alone has the additional power to achieve (2) by impeachment.
