Cops:- they look up Rego plates on computer, so if the old plates (at time of theft) are still on there, I'd change them
Car:- whilst stolen, the thieves may have thrashed the car. You'll get it thoroughly checked over won't you?
History:- VEDA including Revs will always hold that theft history in place. Should you sell the car (after having bought it), any enquirer might have the same questions as you have.