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According to the Swim skill, if you fail your Swim check by 5 or more, you go underwater. There is nothing in the skill that dictates how deep you go.
I'd say becoming paralyzed means an auto failure on your Swim check, which you'd usually be making during your turn, unless some other water hazard presents itself before then.
It's really one of those things that's open to discussion between players and the GM.
I don't think there's a rule for sinking speed, but I imagine it'd be very much like feather fall. Since it typically has a speed decent for a safe landing. So I'd guesstimate 60 ft/round. I'd start the sinking on the creature's turn.
Lucky for it, holding breath isn't actually an action. So it'll be fine paralyzed down there for a few minutes I'd think.
A Giant Turtle is on the surface of the water and is Paralysed.
It has water breathing so it won't start to drown, but it presumably it will start to sink (I'm pretty sure Turtles don't have positive bouyancy).
How fast does it sink, and does it start to sink straight away or at the start of it's turn?
Thanks
It stays in place.
Amongst other things, turtles control their bouyancy by the amount of air in their lungs. This gives them positive bouyancy above the depth they intend to stay on, which unless it has emptied it's lungs just before being paralyzed would include being at the surface.
Some turtles have been recorded to dive to 1000 metres, which would impossible for it to return from, if it had a negative bouyancy at the surface.