What is this spider, and what is this object that emerged from its rear end?
I do not know what species of spider this is, but I think the white ball is actually spider poop.  Unfortunately a quick search did not return many references, but here is a picture for comparison.



Spiders produce uric acid, which is a near-solid and excreted out white.  This is done to minimize water loss. These malpighian tubules drain into an pouch attached to the digestive tract, or stercoral pocket.  The uric acid waste from the is combined and eliminated together with solid waste from the digestive tract.  In your spiders case, this would be undigested insect parts (black) mixed with the uric acid (white).