If your difficulty breathing responds to asthma rescue inhalers, does that guarantee you don't have anxiety and you have a real lung disorder instead?

Not necessarily: I see you have belching, reflux, and now have experienced improved breathing with a rescue inhaler. The improvement with a rescue inhaler indicates you have reversible airway reactivity. This is not a disease of the lungs but, rather, a tendency of the muscles of the airways leading to your lungs to go into a spasm and possibly have inflammation. This can be on an allergic bases as in traditional asthma. In your situation the reflux may be causing small amounts of acid to seep into the airways making them more reactive. Food allergies can trigger increased reflux by causing esophageal inflammation. Having airway reactivity can cause anxiety and having anxiety can worsen airway reactivity as well as increase the severity of reflux. You will need to have the reflux addressed, determine if you have a problem with the ability of food to stay in your stomach and not come back up, see if your belching could be due to Helicobacter Pylori, for example, an infection that can be treated, and determine of you have chronic inflammation in your airways needing treatment.