Hi Gabrielle!
I don't think working out on an empty stomach is ever a good idea. It can leave you feeling faint, dizzy, nauseous, and/or depleted(the shakes). Food fuels your muscles. Now, you don't want to eat too close to your workout time since this will interfere with digestion, but you DO want to have some energy reserves in your system. Eating around two hours before you workout(unless its a very heavy meal)should provide you with some good energy to fuel your workout.
As far as working out in the morning for optimal fat burning, this has still not been scientifically proven to my knowledge. Some research has lead people to believe this but it is still under debate.
Yes, eating 6 small healthy meals is better than eating 3 larger ones. From a health stand point, it is better for and easier on the digestive system. It also helps to keep your sugar levels more balanced. From a body fat loss standpoint, since your sugar levels are more balanced, you will be less likely to develop cravings that can lead to binge eating. Also, with smaller and more frequent meals, you will not be as likely to reach for "in between meal" high calorie junk snacks, which almost always leave a deposit in our fat stores.
Hope this helps!