Can anxiety cause numbness in the lips and mouth?

Numbness. Also called paresthesias, yes it is common, along with other symptoms of anxiety.
Yes. Yes, anxiety can cause numbness in the lips and mouth -- especially when hyperventilating. See your physician for an evaluation of your anxiety, and get help for this.
Yes. If you are hyperventilating, it can. If you breathe into a paper bag, it'll go away. Cognitive-behavioral therapy by psychologist will help you develop strategies to reduce your anxiety, while a psychiatrist helps you with medication management.
Certainly. Numbness in the mouth, scalp, fingers , toes..And other parts of the body can accompany anxiety and are well known and common physical symptoms of anxiety, however, check with your doctor to make sure nothing else is going in and that the numbness is within the context 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.