All three of your destinations should be great in January. Joshua Tree can still be crowded due to European visitors, it's their winter too. Death Valley should be very nice with low attendance. Grand Canyon, if you are lucky enough to see it after a snow, unbelievable in winter. You can still see everything with the added detailing of the snow. It is funny how the snow highlights the detailing in the spires and rocks. Attendance is still low then. Be forewarned that the North Rim will be closed. It closes the end of this month through May 30.
You are well aware, I assume, of what an inversion layer can do to the weather, during January with normally clear skies, it can get pretty cold. Not like AK but 30's to 50's during the daylight hours.