If I have a Dual Nature in play, and then cast a Deranged Hermit, do I get my 4 squirrel tokens for the original Hermit, then 4 more for the "new" token Hermit that comes into play? What about Echo costs on the token hermit? Since as a token his converted mana cost is 0, is his echo 0? Or because he is an "exact copy" is his echo 3GG?

What about with Stangg? Do I get 3 or 4 Stangg?

Net rep commentary issued after the Prophecy FAQ- This (Dual Nature) works very cleanly. It looks for creature cards, not for creatures, so things like Opalescence have no effect. The token becomes an exact copy of the basic card - what you can see on the card is what the token says. You can only copy creature cards, not creature tokens or non-creature cards that come into play as creatures. If it leaves play before the triggered ability to create the token resolves, then the token will stay in play as a normal creature.

Deranged Hermit Color=Green Type=Creature - Elf Cost=3GG UL(R1)
Text(UL+errata): 1/1. Echo ; When ~this~ comes into play, put four 1/1 green Squirrel creature tokens into play. ; All Squirrels get +1/+1. [Oracle 99/05/01]

Principles: Dual Nature is the latest "copy" card to create interesting situations (previously we had cards such as Dance of Many, Echo Chamber, and Stangg all that created tokens that were "copies" of permanents.) Stangg creates a Stangg-Twin (When Stangg comes into play, put a 3/4 red and green Stangg-Twin creature token into play.).
Given that they're copies, I would include all characteristics in the xerox effect, including mana cost. This fits into the "if the card breaks the rule..." breaking the "tokens have mana cost = 0" rule. A key I've noticed, is if the card states characteristics (name, color, P/T), then it's not a copy. All the tokens generating copy cards avoid defining any characteristics.
Thanks to John Carter of the Delphi Group for this explanation

The Answer: - The Hermit token produced by the interaction of playing the Hermit and Dual Nature will have a converted mana cost of 5 and an echo payment due of 3GG.
You will get lots (and lots)of Squirrels.
You will end up with only one Stangg (Legend rule), but a couple of Twins.

What happens when you have a Dual Nature and an Opalescence in play and another global enchantment comes into play? Do you get a copy of the new enchantment? I guess it comes down to the question of what the definition of a creature card is.

The Principles: -As explained in the Comprehensive rules, "A Creature is both a card type and a permanent type." Dual Nature specifies "creature card" to make it clear what will trigger its effect.

The Answer: Enchantments that become creatures through Opalescence's effects will not trigger Dual Nature.