The S-Tronics are mild-hybrids, so require a slight change of mindset: you can pretty much ignore what the engine is doing. Have you tried moving off with the engine stopped? In my car, the car moves instantly, with no lag.
And if you want a slightly quicker pull away, you can switch on AutoHold (in Settings), which will get the car to hold the footbrake for you, allowing you to remove your foot from the pedal whilst stopped, so you can have it on /over the accelerator, ready to go.
But to answer your direct questions. The car fully disengages the clutches when it is held on the brake. And you can switch off Auto Start/Stop using the button with the A on it in front of the gear selector (but I’d advise taking the time to get used to driving a mild hybrid before doing so).