Correct. You cannot take a double move and attack in the middle.
Everyone gets to act the same way in every round. You can either do:
Full Round action
------ or --------
Standard action
All Standard actions include a move -- that is, you are allowed to move your normal speed before, during, or after the standard action. You are allowed to substitute other things for your Move. For example, you can attack (a Standard action) and then Pick up an item on the ground (instead of moving). You can substitute any action listed under the Move Equivalent Action section of page 128 in the Player's Handbook instead of your move.
A double move means you are moving your speed as your Standard action, plus moving your speed as the included Move equivalent action (I don't think this is explained well in the Player's Handbook). If you do this, you can move up to twice your normal speed, but then you don't have any action left to attack.
Making one attack is a Standard action that also allows a move.
In the normal case, you can either move then attack or attack then move. You must do all of your movement either before or after the attack.
If you have Spring Attack, you are allowed to "split up" your Move Equivalent Action into two pieces, one before your Standard action and one after. You are still limited to the normal amount of movement per round (your normal speed), but you just get to split up the movement as I said earlier.
Generally the idea with Spring Attack is that you stay out of reach of your enemies, close to melee range, attack once, and then retreat out of melee range, so that it makes you harder to hit.
It would have been easier if they explained it this way:
Each round, you can do one of the three below:
(1) a Full Round Action,
(2) a Standard Action + a Move Equivalent Action
(3) a Move Equivalent Action + a Move Equivalent Action (this is referred to as a "double move")
The reason you can do (3) is that you may substitute a Move in place of your Standard Action.
What you cannot do is substitute a Standard in place of a move, so you CANNOT do:
a Standard Action + a Standard Action
-------------------
Yes, you can use Spring Attack while jumping. You can jump before the attack, after the attack, or both. You would need a separate Jump check for both unless you are making a single jump and attacking the target as you jump past.
Even if your Jump check gives you a distance greater than your speed, you are still subject to your normal movement limit. (Your movement before the attack plus the movement after the attack cannot be more than your normal speed). If your Jump check allows you to move more than this and you want to move the entire distance, you can cover the extra distance in the next round or rounds by using move actions then. Remember, you can always choose to Jump less than what your Jump check allows.
I believe there is a question in the D&D FAQ at Wizards.com about using Spring Attack while Jumping.
I hope this clears things up instead of making it more complicated.
(edited for clarity)
Jason.