How does the ambience lighting in the convertible turn on? Do you press a button or is there a sensor? (presumably the same one that turns on the headlights)

Perhaps u r confusing ambiance lighting (usually used in nice restaurants for ''mood'' lighting) with ambient light (which is what ever light is present from the environment. Or perhaps u really r referring to ambiance lighting in the real sense of the word.
There is no provision for ambiance lighting in a BMW for most everything is there for functional lighting & not for mood setting.But there is an ambient light sensor built-in to the car which tells the headlights to turn on when it becomes sufficiently dark. To turn on the automatic headlights based on ambient light conditions, the light switch must be turned counterclockwise to the leftmost (position A, as in automatic, & not accessory).
There is also the interior light switch above the rearview mirror, which works with the door microswitches when the door is opened. The remote also triggers the interior lights to come on. The car/key memory which is programmed into the general/basic module also has something to say about which lights come on or not.

Easy.

The ambient light inside the car ( usually a dim orange glow from various small bulbs) is tied into the sidelights/headlights. Basically, when the side/headlights r switched on (either manually or automatically) the interior ambient lights come on as well.

l guess the BMW engineers figured that if it is dark outside the car it will be inside too :)

The sensor for the automatic headlights is on the outside of the windscreen just behind the interior rear view mirror.