Yes. Coffee belongs to a class of drugs called methylxanthines. This drug class also includes "theophylline". These drugs are weak bronchodilators so a good cup of "black" may actually decrease shortness of breath in some patients!
No. Although caffeine is similar to theophylline (a medication used to treat copd) it is not an effective bronchodilator. Theophylline isn't used much in practice since better and safer medications are available, but it is still used in severe cases occasionally.
Not really sure... Coffee does have caffeine, and it is the same chemical class as Theophylline, which is a medicine used for COPD. So, technically, coffee may help COPD. But, there are usually much better treatments than coffee. And...if you drink too much coffee, it can cause other adverse events too. So, drink coffee if you like it (in moderation), but I wouldn't use it to treat COPD. Good luck!