Depends. What happens during the ground check? If you're running the RPM's up, doing mag checks, carb heat checks (if equipped) and prop control checks (if equipped), then yes. If ground check is simply checking that the controls are free and control surfaces are moving correctly, or if its some other kind of check, then no.
As for whether or not you lean again after the runup, again it depends. Generally the runup is performed near the take off end of the runway so I probably wouldn't bother to lean again afterward. But some with some runways at some airports, the runup is done further from the take off runway entrance. In that case the answer again depends. If you lean again to continue taxi, are you 100% sure you'll remember to go mixture rich before full throttle? If so, leaning again is probably ok. If not, staying full rich is probably the better bet. Others will surely correct me if I'm wrong but I doubt you're going to foul anything all that bad in a minute or three of full rich taxiing.