burning calories

You should exercise whenever you can. Definitely do not wait to exercise until you've eaten. You also want to eat after you've exercise to help your body recover.

No.

In general, calorie deficits (eating less than you need) is what makes you lose weight.
Exercise primarily makes you healthy and builds muscle so you LOOK great at whatever weight you are at.

Importantly, exercising also means that the weight you lose is FAT (what you're trying to get rid of) not MUSCLE (which you want to keep).

If you exercise without eating enough your body gets the calories it needs by essentially eating (metabolizing) your muscles. You want it to burn your fat.

Remember also, your body doesn't work day to day. It need enough energy to do those exercises. Then it needs to replenish itself and re-build new muscle and repair injuries with more energy. AND you still need energy to think/breath/walk around, etc.

That energy comes from calories. That's why when you exercise, MFP ups your calorie allowance for the day. You need a deficit to lose weight but not too big a deficit so your body slows down and holds onto calories and you get sick.