Maybe... It depends on the degree of shortness of breath, how far away you are traveling as well as the etiology of your shortness of breath, ie what is causing you to have this? The answer to if you can travel short of breath requires an evaluation by your doctor. Shortness of breath can be life-threatening so you should see your doctor prior to any traveling. And, if your symptoms worsen, go to er.
Depends. You should see a doctor to determine the cause of the problem before your vacation. The problem may be as simple as heartburn or a respiratory infection. If you have a more serious condition or if you are planning on traveling to a remote location at high altitude or to a region with extreme temperatures, you may have to postpone your vacation.