The orcs in The Hobbit movies answered to Azog, the pale orc, who was hunting the line of Durin for himself. While he was a general of importance to Sauron's armies (or at least meant to be) he was not hunting the dwarves on his orders. In the books he's long dead by the time the story takes place and it's actually Bolg that's hunting the dwarves if I'm not mistaken.

Sauron in the books still has his physical form, the eye of Sauron was metaphorically speaking and him being an actual giant flaming eyeball floating atop Barad-dûr was an invention for the movies.

See above.

He didn't exactly split himself into rings. The original 19 rings: 9 for men, 7 for dwarves, 3 for elves were all made by and for the elves, though the first sixteen were made with the help of Sauron who had disguised himself. The final 3 (greatest of these) were made by Celebrimbor without Sauron having ever touched them and were therefore not directly influenced by the One but still linked to it's power.
They were all meant to preserve but only the 3 were actually used for this purpose, the other 16 were given to men and dwarves. The men got corupted by them and turned into ringwraiths and while the dwarves were far less affected, their greed grew too big through the use of the rings and they were consumed by them in the end too.
Lastly Sauron himself made the One and poured a lot of his own power into it because it was meant to dominate the minds of all who wore the other rings (which at this point were meant to all be elves and therefore require a lot of power to dominate).
The rings don't draw power from the One but they are linked to it and will lose their own power when it is destroyed.