I was of the impression that 4e didn't actually have any concept of a "double move" - it was just "one move after another".
You were under the wrong impression then. There are specific rules for a double move, you can search for it in the compendium, don't want to paste the entire text here wholesale.
Actually, I'm pretty sure they allow this by RAW. I'm pretty sure (but not 100%, and don't have my PHB to hand), that there is no Jump action, but rather that jumping is something you do as part of other movement.
This means that you can include a jump as part of a Charge action. And that in turn means that you can Move (including jump) followed by Charge (including jump), to perform that "long charge" you just described, including the attack at the end.
Yes jumping is part of a move action. Say you are speed 5, your enemy is 10 squares away. There is a chasm that starts at 3 squares and ends at 7 squares. You want to charge the enemy. By RAW, you can't. First you have to take a move action, then you have to charge. The move action of 5 squares (part of which is a jump) puts you in mid-air, and you fall. If you were speed 7, you would be able to land at the end of the jump, so you don't end in mid-air. And you would then be able to use your standard action to charge the target now 3 squares away.
You cannot combine a move and a charge into the same action. They are always two separate actions.
In fact, it would even be possible, by RAW, to Charge (including jump) followed by Move (including jump), in order to use a melee attack against an opponent flying in the middle of that chasm.
It's not covered because there's no need. You just Move and then Charge (or Charge and then Move).
Yes you can charge something in midair (I did mention in an earlier post I allow this). But for one, when you end your charge action in midair, you fall. And secondly, a charge action ends your turn, so you cannot take further actions.
You can end a move action in midair and not fall if you can fly.
The only question is whether Jump is a separate action, or if jumping is merely something you do as part of other movement.
Jumping is part of a move action. The distance you jump cannot be split into two move actions, *unless* the are the same move action in which case they can be combined into the same double move action. Move and a charge are two different actions, and cannot be combined.
A charge is a standard action, if your speed is 5 and you move before charging you can absolutely end up 10 squares from where you started which you describe as a "double move attackish thing".
Yes, you can do this during your turn. But this is not a single action. It is two actions. At the end of each action, you must be standnig on solid ground.
As for using this method to jump a canyon, your speed is totally irrelevant since your Athletics check determines how many squares you clear. And yes, I would totally allow someone to make an Athletics check to clear a gap in the middle of a charge if it made sense.
You may allow it. It is not the rules. See my example to delericho above.
To simply say "Ok, you jump and miss. You fall." as though the PC is paralyzed as he falls and is unable to do anything else until he hits the ground, especially when he has the magical physics-defying ability to instantly move from point A to point B without crossing the space in between is so terribly bizarre.
If he could use that ability as a free action (such as with Feyleaf Sandals), then sure he could save himself from falling. But it is a standard action, so no.
If there were 5 dire bats in midair over a chasm, and the first level eladrin monk wanted to jump up in the air over the chasm, then use Masterful Spiral to attack all of the dire bats, then action point and feystep onto solid ground, would you allow that? I mean sure, it sounds awesome, but if you could do this, what would be the point of flying?
You cannot take a standard action in midair, unless you're flying, or some other power allows you to do so.
Just because it isn't explicitly written in the rules doesn't mean it can't be done.
In matters not covered by the rules, sure. The rules are a framework for conflict resolution. If I can find an answer to a conflict in that framework, I will use it. If not, then I'll come up with another solution. I realize my opinion differs from many others here, but being awesome is not a license to ignore the rules.