A serious Republican House leadership could force the appointment of a Special Prosecutor, although I think the current terminology is Special Counsel.
I'd question whether shutting down the government would force this administration to appoint a Special Counsel.
They won't brook anything that might inhibit their lawlessness.
Yes, they could but seeings how there is no ‘special prosecutor’ provision anymore (it was not renewed IIRC) an appointed special prosecutor wouldn’t have much influence.