You are correct in that it's not definitive. It says Move your speed as part of the charge and make an attack at the end. If asked I would say yes you can Run (+2 movement speed) as part of a Charge but you'd suffer the normal penalties for running - giving combat advantage until your next turn, -5 to hit. Heck technically I'd say it should almost be a requirement to run during a charge. Hard to picture a 'charge' at a walking speed.
The purpose of the charge is, IMO, merely a way to give the combatant a way to take two movement actions and still attack in the same turn without having to spend an action point. Turning both movements into a run to get +4 movement and taking the -5 to hit just lets you move farther and still have a chance albeit 20% less to get that extra little something out of a round in terms of distance and damage.
#1 makes me giggle.

I would say that unless you are running, you are advancing rather than charging.
Right. That's the obvious, common sense answer. However, by the rules as written, you can make a legitimate argument that you cannot run while charging... "move your speed" is not the same as "take a movement action".
In the case of 2, I'd say you'd be breaking the superhero code if you don't fly while prone, flying standing up, unless you're on a magic carpet or something similar is just gauche beyond all limits.
#2 It depends on why the character is prone? If the character is not suffering any condition that prohibits movement and is merely lying down then I don't see why not. A winged creature lying on its back might be limited to a half move on a move action due to having to roll over but thats just one possible way to judge an initial move from that position.
Okay... forget visuals for a moment. By the rules:
Description aside, prone is just a condition that grants certain penalties and benefits (see
PHB page 277). Now, there are two move actions directly related to being prone: Crawl and Stand Up. However, there is nothing I can find that specifically says that you cannot take other types of movement actions while prone.
However, common sense says you must stand up, before you can walk or run.
In addition, though, Prone does specify that a prone flying creature falls. In other words, regardless of the posture it takes while flying, a flying creature that is "knocked prone" effectively stops flying temporarily.
So, consider the following situation:
An Adult Green Dragon Dragon is on the ground (not flying), dazed and prone. She's got only a single standard action on her turn. By the rules, can she use her flyby attack which, as a Standard action, allows her to fly up 12 squares and make a bite attack at any point during the move? Or must she stand up first?