You will digest more efficiently during sleep as your parasympathetic nervous system is dominant at rest which favours blood flow through your enteric bloodstream which enhances absorption of gut breakdown products. Parasympathetic activation also stimulates salivary gland secretion, gastric activity and peristalsis.

During exercise, the sympathetic nervous system is predominantly active and the release of catecholamines stimulate peripheral vasoconstriction to the renal and mesenteric circulation effectively reducing blood flow to those systems whilst enhancing blood flow through the coronary and skeletal muscles system in the fight and flight response.

1) When you are upright, digestion is aided by gravity to some extent, even if you don't have acid reflux. So that's one disadvantage to digestion while asleep, unless you sleep sitting up.

2) Beyond the question of how much digestion happens at night is the question: how much digestion SHOULD happen at night? Digestion diverts blood and oxygen away from your heart and brain. When you sleep, your breathing and cardiac output are reduced. So if you eat a large meal before sleeping, too much of your body's resources could potentially be diverted to digestion, which could affect the restfulness of your sleep. Since early morning is also the highest risk time of day for heart attacks 

it has been shown rather conclusively that, in terms of weight gain/loss, it doesn't matter when you consume food
would have to disagree with Paleo Dave on this occasion. The success of intermittent fasting (IF) strategies would argue that it DOES matter when you consume food. You can burn more fat by spacing out your feedings. IF favorably affects your metabolism and hormonal milieu. If you eat every three hours, your body will not have a chance to tap into fat stores. That being said, I don't know that it matters what time of day you do IF