The entire impeachment process occurs in the House; but once that is complete and- assuming the person is actually impeached- the process then moves to the Senate, where the trial is conducted to determine if a removal from office is warranted.
The impeachment (and removal) will take precedent over pretty much any other business that the Senate may have. That's kinda what I mean when I say that the House- if it really, really wanted to- could jam up this whole process.
But if that happened they should really just tear down the entire structure because it would be a shitfight of epic proportions.
