Are creatures or objects in the water and within 25′ of the vortex automatically pulled 10′ toward it?
Yes, when the whirlpool forms.
Do they continue to be pulled 10′ further toward the center of the vortex every round they are in it?
You’ve pointed out that control water allows the caster to spend an action to “repeat the same effect or choose a different one” — so the caster should be able to spend an action on their turn to re-form the whirlpool and pull creatures ten feet toward it again. This seems like not the best use of an action but it appears to be a valid use of the spell.
If the caster does not choose to use their action in this way, then the pull does not repeat.
Is a creature or object caught in the vortex also pulled down toward the bottom?
It only pulls “toward” the vortex. We don’t have a clear definition of what it means to be pulled “toward” something that is larger than five feet, so your DM will have to fill in the details of how this works. This DM would rule that you stay on the surface of the water, but the surface of the water is sloped downward towards the center, so if you reach the center then you are still on the surface of the water but you are twenty-five feet lower than the normal water level.
On a failed save, are they caught for the entire duration of the spell, with no more chance at saves, or until they can swim free of it?
Every time they start their turn in the vortex, they make a new save, and the result of the new save determines whether they’re “caught” during that round. The fact that they’re “caught” seems to only affect whether they get disadvantage on a check to swim away.
This is a bit confusing because failing a save causes a creature to be “caught in the vortex until the spell ends”, but then making a save on the next turn overrides that by saying that the creature “isn’t caught in the vortex”.
On a failed save, does a creature continue to take 2d8 bludgeoning damage every round they are in the vortex?
The creature makes a new save every round they are in the vortex, and takes full damage or half damage depending on the result of that save.
Can the whirlpool be cast so that it is completely underwater?
The rules don’t technically say that you can’t do this, but your DM might veto it.
Also, can the whirlpool be cast ‘upside down’ so the widest part is at the bottom?
No. The rules say “the widest part is at the top”.