How does the post-2.0 Puzzle Ring work?

I confirm the following after after some testing:

  • The ring does not work if equipped by your follower
  • The goblin is summoned when equipping the ring and follows you around as long as the ring stays equipped, unequipping it will reset the item counter
  • Dying will reset the item pickup counter, which means you must start over with collecting normal-quality items after you die
  • Exiting a game resets the counter
  • Exiting an Act also resets the counter
  • The goblin will not pick up player dropped normal-quality items nor will it pick up normal quality crafting mats
  • The goblin will pick up low quality item as well
  • It seems that when you go from zone to zone, the game unequips and re-equips your items

Big areas to farm the normal/low-quality items:

  • The Cursed Hold (Act 1)
  • Halls of Agony (Act 1)
  • Battlefields of Eternity (Act 5)

Legendary drop rate is currently unknown.

Patch 2.0.5 notes the following changes:

  • Legendary drop chance doubled
  • Rare items dropped will now always have six affixes
  • The counter for picking up normal-quality items will now persist through death, act transitions, and un-equipping the item (Note: This still will not persist across game sessions)
  • The goblin will now play a sound when its counter reaches max and it drops an item

Even though I can not answer all of your questions since I did not get a puzzle ring twice yet, I can answer some of them.

  • There is no cooldown on the goblin. As soon as you equip the ring, the goblin appears next to you. As long as you stand shortly next to a normal quality item (in my experience: even a low quality item), the goblin will pick it up. At this point, it "disappears" (it does not "pick it up for you" as the description says, but rather takes it). After the goblin has picked up enough items, it drops the mentioned items.
  • In my experience, the level of the item always matches that of other loot drops (equivalent loot drop).
  • As far as i know, the drops of the treasure goblin work like any other loot drop. When determined which type of drop is going to happen, the magic find as well as other modifiers will be taken into account.

This answer is only based on my experience though, not on any type of datamining etc.