Can anxiety cause dizziness, lightheaded-ness? Feeling kind of faint? Can a cold also bring on some of those feelings?

Yes, multiple causes. Anxiety can cause a multitude of physical symptoms, including those you've described. A cold generally starts with other kinds of symptoms like stuffy nose and sore throat, but if the congestion gets into your ears, dizziness ; lightheadedness can result. Many other possibilities as well, including xanax (alprazolam) use. Hoping you're working with a psychiatrist on your anxiety issues that you listed also.