Given that you're a beginning runner, chances are you won't be exercising so hard that you'd need additional energy before working out. So you should be able to go out running without breakfast.
For building up stamina, there's no real way to 'rush' the process, because your body will need time to adapt itself and rushing ahead of your body is only likely to get your injured. Instead, you should focus on a training program that's fun, easy to keep up and won't get you injured. That way you'll be able to slowly, but surely built up your fitness and make long term changes in your lifestyle. Also important to note is that the calories you will be burning the first few months are negligible, but over time you'll progress and increase your workout frequency and intensity. That's the real goal: getting fit enough for the worthwhile exercise. The catch is, you need to stick to exercising long enough to get to this level of fitness.
So Instead of trying to run non-stop without any structure, I'm going to recommend you try the C25K or if you have a smartphone the Adidas micoach app. They both offer training programs for absolute beginners with no stamina, but if you strictly follow the program, you should be able to run non-stop for 5k within 10 weeks. An important thing to note though, is that you shouldn't be running as fast as possible, but instead 'run' at a speed that you're still able to hold up a conversation. You'll be surprised how slow this pace may be, but its probably the single most effective way of preventing injuries.
As for flexibility, I've been quite vocal about stretching being quite controversial, but a blog post I found recently called Research Says Don't Bother With Stretching Before You Run mentions that more important than stretching or not was:
switching pre-run stretching routines (runners who normally stretch stopping and those who do not stretch starting to stretch before running).
So if you never stretched, starting to incorporate this into your pre-run routine is a pretty bad idea. Instead, if you feel that you have a limited range of motion, start stretching regularly and in a more functional way such as yoga instead of focusing on a arbitrary set of muscles.
Good luck with getting started and be sure to come back if you have any follow-up question as you progress!