A: Obviously, it depends on the circumstances. Did you provide consent? Was the vehicle moved only a short distance?
An officer should not want to drive your vehicle unless it presents a danger to others i.e. to protect others, prevent imminent harm, etc. If your car is parked in an intersection and you are impaired then the officer could move the vehicle to the side of the road and out of the intersection in the interest of public safety.
The normal practice would be for them to impound your vehicle if they stop you and have a warrant to serve you. Part of their logic is that it gives them an opportunity to search your vehicle under the guise of "taking an inventory" as part of the impounding.
If they did not discover evidence that led to a charge, but they took your vehicle to your residence or allowed you to take it to your residence, prior to taking you into custody that should not be a disservice as it saves you impound costs.