Excellent questions! I will tackle the easy one first. A dermatological response to an allergen might be hives, or rash, redness, itchiness, or swelling.
No, a chest xray will not be sufficient to detect cardiac problems, more appropriate tests would be an EKG, Echodoppler ( or Echocardiogram), or in certain circumstances, TEE or transesophageal echocardiogram. And yes, sometimes a heart condition can cause a cough.
Singulair belongs to a class of drugs which inhibit leukotrienes - part of a allergy response by the body. It works best when taken routinely and faithfully. It does not work well as a rescue drug. Some of the other meds you mentioned trying also help against a histaminic response, but your original post stated that you do not have nasal stuffiness, which some of the other meds are useful for. This may explain why you did not find them tremendously helpful. The manufacturer of Singulair is Merck Pharmaceutical. As a company, they are available to answer questions about their products. You may wish to contact them to see what they can tell you about the use of Singulair for you.
Congratulations on being a non-smoker!