AFAIK, the mini is an electronically shifted automatic - like the vast majority of them. If you are asking "like the BMW", you could be implying the SMG box on the M3. That's a sequential shift manual box with electric shift - similar to what is on a Ferrari (or F1 car). Don't know about DSG, but if it's on a beetle, I'm guessing regular stuff.
Paddle shifters, tiptronic, steptronic, multitronic, trontronic are ways that the driver can control the shift point on the slushbox. And slushboxes continue to get better and better and better. So for the majority of street driving, and in particular, LA stop and go, a slushbox is a wonderful thing.
In Beth's XC90, there is a manual mode to the auto trans. Yeah, lots of time rowing through the gears in the big AWD SUV! But it is handy for hill descents.