How are calories processed after eating?

I have no scientific explanation for why it is sometimes harder to workout in the morning. For me, it has always been a hydration issue if I'm having trouble exercising in the morning. Drinking plenty before working out in the morning has helped.

Energy is a complex combination of many sources (fat from fat cells, glycogen from liver and muscles, protein in muscles, blood glucose, other body tissue, ketones..etc). You usually use multiple sources of energy with a bigger percentage of one source over another depending on the situation, and the body even stores and burns energy at the same time. If the burned energy is more than the stored you lose weight.

At rest you are predominantly burning fat. When you do cardio, the higher your heart rate the more carbs you are burning because, though not as concentrated per unit of weight as fat, carbs need less oxygen to be converted to energy. If you do a long and/or hard workout in the morning on an empty stomach you can potentially deplete a lot of these stored carbs so your performance can suffer while your body is trying to use more of the slower sources of energy like proteins, that's why it can feel harder. If you feel you do best exercising after eating then do that. It really doesn't matter which source your body burns predominantly because if you are eating fewer calories than your body burns everything balances out and you end up burning fat in all cases.

How long does it take for food to convert to energy? It depends on many factors like the type of macronutrients it has, the type of food, the volume of food, what you are doing, the levels of your hormones, absorption rate, temperature, what your diet looks like overall, your unique physiology...etc, so it's hard to give an exact answer. Even one food can give you some of its energy sooner and the rest of it later.

The only thing you need to know, really, is that excess calories lead to weight gain and a calorie deficit leads to fat loss. Any other micro-management stuff like fasted cardio, meal timing..etc are not really that important or necessary for the average dieter.

"No pain no gain" basically means you need to put in some effort to get results, can't do it by randomly wishing it without making some changes. It doesn't mean you need to deliberately make yourself tired to get results. A workout after a sleepless night feels harder, doesn't mean it burns more fat than a workout with adequate sleep. Your perceived exertion is not a measure for the amount of fat burned. So feel free to eat your breakfast if it helps you perform better and don't focus on minute details that have near negligible effects on your weight loss.