You have no real handle on the scale of damage being done and DOGE is a real department as it was merged into the US Digital Service through executive order.
No, USDS is an organization within the President's executive staff.
The constitutional requirement is that "Officers of the United States" need Senate confirmation unless Congress has provided otherwise. The precise contours of this have never been super well defined, but it doesn't sound like Musk is exercising sovereign power under his own authority, at least not yet.