Depriming with a universal depriming die allows you to clean the cases before sizing and reduces the chances of scratching your dies and brass.
Absolutely correct! Clean the case thoroughly and your sizing etc. goes smoother from there on, and may in fact provide a more precise job of sizing each case to the same external measurements without dirt etc. interfering. And from experience, you'll spend a lot less time cleaning your dies.
My method - deprime fired cases in a universal depriming die, rinse thoroughly in hot water, then run them through my Hornady liquid case cleaner using boiling water and the Hornady proprietary case cleaner. The sonic vibration really does work as long as you make sure each case is filled with the solution, though I have to admit it takes 3-5 cycles to get them fully clean. AND BTW - don't get crazy with the Hornady cleaning solution...too much and the cases seem to end up discolored.
A couple more hints - if the cases have been sitting for awhile (like days/weeks etc.) let them soak in a solution of hot water and the Hornady solvent for 3-4 hours and THEN run them though the sonic case cleaner. Once they are clean, simplest way to dry them is on a tin pie plate in the oven for about an hour at 160-170 degrees. Note that liquid cleaning/drying can leave watermarks on the cases, but nothing an hour or two in the tumbler with corn cob or walnut shell media doesn't get rid of. Not a bad idea anyway if you like your cases shiny!