Diet is obviously the most important, but a small short run just doesn't burn calories. It will short increase your heart rate and metabolism, but eventually your heart rate returns to normal as does your metabolism.
As for cutting calories, you have to be extremely careful. If you cut too many your body will do into survival mode and you will stop losing pounds after just a couple of days. It will essentially stop burning calories to perform simple non essential needs. You'll lack energy and feel like crap. A good mix is the best. Basic workouts can also significantly burn calories and help shed unwanted fat. You just have to stay active. 1 hour in the gym where you lift 15 minutes worth of weights isn't going to help you. But a crossfit style workout or some other form of workout where you are continuously going, working out several different areas will boost the metabolism and help burn more calories.
This is something that is often said, never backed up with sources, and almost always wrong. The study that this information comes from found that you will go into starvation mode if you're restricting your calories, while at 5-6% bodyfat. That's professional bodybuilder PEAK bodyfat percentage, so 99% percent of people do not need to worry about going into "starvation mode".
The problem is that the calories you take in at an extreme deficit need to be VERY well thought out. People feel like crap with no energy while in a deficit because their calories are made up of rice cakes and lowfat yogurt. What kind of diet is that? It's a pretty ****ty one with almost none of the proper nutrition that your body requires. At low calorie levels, most of your calories should be coming from protein.
I've been going at a 1000 calorie deficit for a few months now and I have more energy than anyone I know. I run everyday but Sunday, I do calisthenics three times a week, and now that it's getting nice outside I'll be outdoors every weekend. That being said, my diet is pretty clean. Breakfast is a fruit smoothie or pita/hummus along with a multi-vitamin, I have granola and grapes for a snack, a tv dinner for lunch (this is my one downfall), and orange for a mid-afternoon pre-run snack, a clif bar or more pita/hummus post-run, and lean meat and veggies for dinner.
I'm also cautious to recommend cross-fit to anyone. Most people have no clue about the fundamentals of proper form on regular barbell exercises, let alone combining power cleans, front squats, military press, and back squats into ONE exercise. It would be very easy to hurt yourself if you don't know what you're doing.
