I'm not sure if I'm yet good enough to implement this, though I can try it. Although it looks like Box2D can do exactly that, as per Zael's post?
Edit:
I think I managed it
I used, like suggested by Zael, Box2D for collision detection and SFML for rendering the graphics. I'm still getting a stack overflow with more than 1700 objects, but everthing below that works perfectly fine. Thanks again for the help!
Edit 2:
The stack overflow is caused due to a too big array. I switched over to a pointer, which solved the problem.

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