Numbness. Also called paresthesias, yes it is common, along with other symptoms of anxiety.
Yes. It is not uncommon for psychological anxiety to lead to physical symptoms (and vice versa). One such physical experience includes sensory changes around the mouth and in the extremities (eg fingers) . This occurs as result of deep/rapid breathing (which can occur when people are anxious). That being said, consider discussing your numbness with your physician especially if it is a new experience.