I've written a good deal on the duel here before, which you can find indexed here. This one is perhaps the most comprehensive. That should cover a lot, but to give you the quick rundown of your specific questions:

How did dueling differ between, say, Austria and the United States?

There are several strains of dueling 'traditions'. The 'Early Modern' kind of stands alone, and then when we look at the 19th-20th centuries, you can roughly break down to the Anglo-American, Franco-Italian, Germanic, and Russian styles. It's a rough breakdown, and there are overlaps in places, and some differences within them, but it works in a pinch.

Did it make a difference (in general) if the duel occurred in an aristocratic society vs a more "democratic" one? (Like in the difference between Russia and the UK)

Yes, the difference in political systems influence how the duel was understood, such as in 19th-c. France where it was a symbol of Republicanism, while in Germany during the same time it lacked that same political impact.

Also, did the location of the duelants influence their preferred weapons?

Again, yes. The Franco-Italians prefered the sword, while all the rest were generally inclined to the pistol. In America specifically you would see an array of weaponry, such as rifles or shotguns.

Much of that is all covered in the linked answers if you want more expansion, but if you still have follow-ups, do let me know.