How does Acrobatic Charge (such as the Swashbuckler ability) apply then? Do the Rules Compendium entries make Acrobatic Charge moot, or does it work differently?
A swashbuckler is automatically able to do those things, no skill checks required.
Acrobatic Charge (Ex): A swashbuckler of 7th level
or higher can charge in situations where others cannot.
She may charge over difficult terrain that normally slows
movement...
This would require a jump check to avoid.
... or allies blocking her path.
Tumble wouldn't allow this, only jump allows you to avoid obstacles. You can't high jump, only long jump. To get enough vertical height on a long jump to leap over a 5-foot character is a DC 20 (or higher) Jump check.
This ability enables
her to run down steep stairs, leap down from a balcony,
These are probably now doable via Jump checks, according to Rules Compendium.
or to tumble over tables to get to her target.
Tumble won't let you ignore such an obstacle, because the Rules Compendium stipulates you must "continue to meet all other criteria for making a charge before, during, and after tumbling." Your path is still obstructed. You could jump over it, though.
Actually, the Tumble skill says that "
obstructed or otherwise treacherous surfaces... are tough to tumble through." Not impossible. And if you can tumble through a square occupied by an enemy, why couldn't you do the same through an ally's square?
Also, you failed to mention the last part of the Acrobatic Charge description:
Depending on the circumstance, she may still need to make appropriate checks (Jump or Tumble checks, in particular) to successfully move over the terrain.
So no, a Swashbuckler isn't automatically able to do those things. They still have to make the appropriate checks.
The Rules Compendium only states "as long as you continue to meet all other criteria for making a charge before, during, and after" you jump or tumble. I think I understand. The criteria is either A) you cannot charge over difficult terrain or obstacles as stated in the PHB, so no Jump or Tumble checks (unless there is no difficult terrain or obstacles, but then you wouldn't need to make the checks), unless B) you have an ability, such as Acrobatic Charge, that allows you to do so. I could be wrong, but that seems to make sense to me.