What is the recommended time for running, in respect to meal time and hungriness (amount of sugar on your blood)? If the answer depends on the goal of the exercise, please explain for weight-loss.
In general, I don't think that whether you're hungry or not is relevant at all.
Just because you're hungry doesn't mean that you don't have sufficient glycogen stores for a run. If you are hungry, the hunger will probably subside once you start running as exercise causes you to have an increase in insulin. Any increase in insulin is followed by a low, at which point you'll feel the hunger return. This may be after you're run, or if you're exercising for a longer period, it could be before your workout is over.
Cardio for weight loss is often suggested to be done first thing in the morning on an empty stomach, that way you will start burning stored glycogen immediately, and if you're eating a low carb diet, possibly go directly to burning fat for energy. Again, whether you're hungry or not is still irrelevant.
If you just ate, you will burn sugar in the bloodstream first when you run. Depending on your metabolism, and how long since your last meal, you might not even digest that last meal yet because blood will be pulled to the extremities during your run, and away from the stomach in order to supply oxygen to the muscles working during exercise.
I usually run after work, but before dinner. I definitely notice that I have less energy than when I haven't had an afternoon snack. I have often run when I was feeling hungry, but rarely feel light-headed after running. I'm usually hungrier after running. I think I do my best when I run about 1.5 hours after eating. That's long enough to not be nauseous, but have a good amount of energy from the meal. I find that my schedule and the weather control when I run more than anything else.
The 'burning more fat on an empty stomach' thing is mostly bullshit. Weight change is pretty much determined by the difference between your calorie intake and calories spent. Whether you're hungry doesn't factor into how effective an exercise is at burning calories.
I always try to eat something an hour or two before working out, to avoid dizziness and nausea from my blood sugar dropping too low.
As for weight loss, you're better off working on that in the kitchen. To get in shape, I'd shoot for weight lifting instead of just running.
I often find that I'm hungry when I first wake up. However, a few minutes into my run I no longer feel hungry. I'm assuming the insulin increase is responsible for this. I have also noticed that the mornings when I run (or do calisthenics) I stay "un-hungry" for up to a half-hour or even an hour after I work out.
I think (hope?) that this is because my body has turned to burning fat for fuel once my glycogen stores are used up, because there's nothing in my stomach for it to digest, but I don't have a source for this.