Engine revving on its own in first gear

I recently got the car listed above with1.2L TSI 90 DSG 7-speed, at 29,000 miles, and have some questions about the way it drives / if it's normal or not. It's really three questions, but the first one worries me most.

1. The car is at a standstill, I go off the brakes, and it's in first gear, as it should. It seems to be at biting point but doesn't move, so I push the accelerator lightly to go forwards, then come off it (because I want to brake, e.G, when I'm parking in a small space), and the engine suddenly starts revving up and down by itself for about 7 seconds before calming down, all this while the car doesn't move anywhere. It does this only for the first 5 minutes of driving, then the problem disappears. What's going on here?

2. I accelerate and as the DSG shifts from first to second gear, there is a huge jolt / lurch forward, but only for the first 10 minutes of driving, then it smoothens out. It's much more noticeable if I accelerate slowly, as if the gear shift is too premature. Is this normal? It feels really horrible.

3. In all the DSG gear shifts during acceleration, the rev counter rises up then falls on a gear change. But not in the second-third gear shift. The engine suddenly revs up for half a second (and you can hear it), as if not having engaged the clutch properly, then quickly engages and returns to normal. Is this a clutch slip between second and third gear?