I get a dry cough every winter, does not produce mucus and a metallic taste in my mouth, labored and wheezing breath which worsens during exercise

Sounds like asthma: You likely have bronchial asthma which is often worsened by breathing in cold and dry air. Interestingly dust mites population usually increases from early winter to early spring and thus you may have a dust mite allergy as well. You need to get a lung function test and may need a controller drug for asthma at least during the winter months. The metallic taste s-> ? sinusitis or gum disease.