Can doxazosin cause dizziness and shortness of breath?

Yes. More common dizziness or lightheadedness less common blurred vision confusion dizziness, faintness, or lightheadedness when getting up from a lying or sitting position fainting (sudden) fast and pounding heartbeat irregular heartbeat shortness of breath sweating swelling of feet or lower legs.
Sure can. Doxazosin can lower your blood pressure significantly causing dizziness and shortness of breath. This can happen even after you've been on it a long time, though it's more common when you first start it. Side effects are not common, but they definitely can occur.