> Can you paddle up a gear without releasing the throttle ?
Yes.
> Does it cut the engine automatically ?
Don't think the engine cuts, I think it just relies on the speed of the change. Given that the box starts to slip with more torque makes me think that is probably the case.
> Can you use the stick instead of the paddles ?
If you move the stick to the right it goes into manual mode and the stick functions like a sequential shifter. You can still use the paddles, but you can push the stick up to go down a gear and back to go up one.
> And if so, how do down shifts work ?
Push it the stick forward if in manual mode and\or pull the down paddle. You'll probably be braking when going down gears anyway and you'll enjoy how smoothly it goes down the box, keeping the car very poised.
> Also, what are the gearbox options and traction control options and how
> do they combine ?
Gearbox in auto mode can be normal, sport and supersport. You'll only really use it in normal when in auto mode. As well as speed of gear change the mode also dictates how sensitive the throttle is (in manual or auto). In auto mode obviously the mode dictates when the gears change and what gear is held at what speed. The box has no notion of "cruising" though. If you're steady on the throttle it'll still hold a low gear\high revs which is why the sport modes aren't suitable for normal driving, and for "proper" driving you'll be in manual mode anyway.
TC is "on", "kind of on" and "almost off". On will keep the car stuck to the steering line as much as possible. "Kind of on" will allow the car to get more out of shape, allowing some slide, and "almost off" gives you even more leeway in terms of steering vs direction. There is quite a good video on YouTube showing the different options. I doubt there is any connection between the gearbox mode and the TC.