The Industrial District is a completely accessible part of Arkham City throughout the whole game. However I'd suggest exploring it only after familiarizing yourself with the game and unblocking some new skills and gadgets, as you will constantly come across enemies equipped with firearms.
There are a couple building that you can enter in the Industrial District, but only two of them are worth mentioning. The first one is the Storage Warehouse 5B (screen above), you can find how to get inside it in the main guide, to be precise in the Heart of Ice walkthrough.
The second building is Zsasz's hideout. Just like above, I'd suggest checking out the main guide, the solution of the Cold Call Killer side mission to be precise.
Batman Trophies - 37
Riddles - 9
TYGER cameras - 12 (4 challenges) + 1 camera Relay HUB
Balloons - 12
Catwoman Trophies - 5