Full-Round Action
A full-round action consumes all your effort during a round. The only movement you can take during a full-round action is a 5-foot step before, during, or after the action. You can also perform free actions (see below).
Quick Draw [General]
Prerequisite
Base attack bonus +1.
Benefit
You can draw a weapon as a free action instead of as a move action. You can draw a hidden weapon (see the Sleight of Hand skill) as a move action.
A character who has selected this feat may throw weapons at his full normal rate of attacks (much like a character with a bow).
Normal
Without this feat, you may draw a weapon as a move action, or (if your base attack bonus is +1 or higher) as a free action as part of movement. Without this feat, you can draw a hidden weapon as a standard action.
Which is why I ask, why "no technically", why no at all?
That's why I said Yes to #3.
So, you could technically say yes to drawing it (without the feat) as part of the move on the charge, though I wouldn't.
This is why "technically" for a charge (which was the original question and what I was answering):
If you are able to take only a standard action or a move action on your turn, you can still charge, but you are only allowed to move up to your speed (instead of up to double your speed). You can’t use this option unless you are restricted to taking only a standard action or move action on your turn.
It says you are limited to only a standard action or a move action. The PHB also cites during the surprise round - which technically allows free actions (if the DM allows) - so that could be a cited exception to the times you can charge with limited number of actions.
Before someone brings up speaking - remember that speaking is a specific exception to free action rules since it allows you to speak even when it isn't your turn in the initiative order.
I wish I had access to my Rules Compendium to see if there is more written in there on the subject but I am only going by the PHB (and SRD) at the moment.