You can take pictures on the tour of the stables in certain parts of it, like the tack room, but not of the horses when you're inside where you are close to the horses stabled there. There is no place where you can photograph the horses unless you get lucky--I did--a handful of riders going from--maybe an arena to the stables or back area, not while I was on a tour but on my way to pick up my ticket. But there is not really a place where you can take good photos. They're strict about it.

The tour doesn't have a performance. The morning exercise sesions are just that exercise there isn't a performance and none of the jumps etc. However on a weekend you can see a practice performance but check their website just to be sure.

Ja, das ist notwendig. Sie kriegen danach Originalkarten beim Kartenschalter im Hofburg

Not far at all. Most places walkable.

Not inside the arena. But you can in stables.

Photography was allowed on the guided tour.