Depending on state regulations.. some states unless handicapped, allow no shooting for any reason, be it hunting, target shooting, anything.. of or from a motorized vehicle. Whether that vehicle is moving or not. So check your State Regulations for that one.
As for the .223 that is a great varmint rifle. I have a friend that used to shoot a Ruger Mini 14 in .223 and he would coyote hunt with that. Also deer hunt. But he said it worked really good. I always used a 7mm mag. But you better not want the fur then.
Sounds like a great birthday present. It is hard to beat a Remington 700 for accuracy. I have been tempted to get a tactical rifle in .223 just for the fun of shooting it. But you could varmint hunt with that as well.