Let's start with the general case and some definitions.
Action: The check is usually part of a move action, but it can be part of any of the creature’s actions that involve the creature moving.
Therefore, in the general case you may absolutely jump as part of any action that involves moving, like a charge.
Looking at your scenarios, Situation 1 is true. You can jump as part of a move. Situation 2 is true, given that there is a square before the enemy, you may jump as part of the implicit move in the charge.
Situation 3 is fun. Case 3a presumes no magical assistance. We consult the following:
If the creature runs out of movement before landing, it also falls. However, if the jump was part of a move action, the creature can continue the jump as part of a double move, ending the first move action in midair and continuing the jump as part of the second move action. The creature makes a single Athletics check for the jump but can use squares of movement from both actions for it.
Which then simply requires us to satisfy ourselves as to the text of the Charge action:
... Move: The creature moves up to its speed toward the target. Each square of movement must bring the creature closer to the target, and the creature must end the move at least 2 squares away from its starting position.
Therefore, there is no implicit "move action" as subset of charge, the movement is an effect of the use of the power.
Looking at Double Move:
Same Move Action Twice: To take a double move, a creature must take the same move action twice in a row on the same turn—two walks, two runs, two shifts, or two crawls.
Therefore, the subset of move action that characters normally employ, the walk:
- Action: Move action.
- Movement: The creature moves up to its speed.
is itself a type of action that comprises a type of movement. A charge is not a walk, and therefore you cannot walk/charge such that you jump on the walk and land in the middle of the charge.
Therefore, if you have to spend a round in the air between an evil guy and a pit, you fall.
You may want to avoid this. Alternatively, grab pouncing armor which, provided a sufficiently high athletics check, allows for jump-charges that exceed your base movement.
Your situation for "needing to charge the guy on the other side of a large pit" is... assuming you must charge them, instead of getting your portable arty to rain death on them and you keep off minions on this side, take a double-move to jump the chasm, then action-point charge. Just... don't fail the jump.