Can you develop asthma when you are older?

Yes. It is possible to develop asthma in adulthood. Allergies may or may not be a trigger. Here is a really good link: http://www.Aafa.Org/display.Cfm?Id=8&sub=16&cont=36.
Yes. Yes. If symptoms can be reversed back to normal it may be called asthma. In adults asthma is suspicious for something else like "cardiac asthma or cardiac disease", emphysema, or reactive airway to something ( allergens).
Not common. Depends on what "older" means. Asthma usually begins in childhood, but can effect people of all ages. Especially in adults another cause of similar symptoms should be considered - alpha-1 antitrypsin deficiency. Also, asthma and emphysema (COPD) are similar in symptoms but very different in treatment, so the diagnosis needs to be clearly made.
Yes. A person can develop asthma for the first time at any age. So, if you develop wheezing, cough, shortness of breath, see a doctor for evaluation and treatment. Treatment for asthma can lead to long intervals where the patient is symptom-free. If you develop acute shortness of breath, go to an er as there are many disorders that can cause this and it can be life-threatening.