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I have a  {Possibility Storm} in play and i cast a  {Grapeshot}. {Possibility Storm} does it's thing and i get another {Grapeshot}. What happens during the initial cast, then the second cast?

You don't get a storm counter for the first grapeshot to copy with unless you played a spell before it.... So you cast grapeshot. Exile it and reveal happening to hit another grapeshot. The new grapeshot storms for whatever your storm counter is. Which if you had only cast grapeshot would be 1. Which is only 3 damage. 1 for the grapeshot from hand and 2 for the effect revealed grapeshot plus it's storm of 1.

If you had played something else before the grapeshot from hand your storm counter would be higher.

Actually the caster of  {Spelltwine} chooses the order. Here are the full gatherer rulings:

7/1/2012   Spelltwine has two targets: the instant or sorcery card in your graveyard and the one in an opponent's graveyard. You can't cast Spelltwine unless you can choose both legal targets.
7/1/2012   If one of Spelltwine's targets is illegal when Spelltwine tries to resolve, you'll still exile and copy the remaining legal target.
7/1/2012   The copies are created in exile and cast from exile. The cards remain exiled.
7/1/2012   You can cast the copies in either order.
7/1/2012   The copy you cast first will already be on the stack when you cast the other copy. If that spell targets a spell, you may choose the first copy as a legal target.
7/1/2012   If you cast a card "without paying its mana cost," you can't pay any alternative costs. You can pay additional costs such as kicker costs. If the card has mandatory additional costs, you must pay those.
7/1/2012   If the card has X in its mana cost, you must choose 0 as its value.

The key question here is can he legally choose to cast the dispel first? The reason for that being Spelltwine states "cast the copies if able" the only legal way to satisfy that statement is to cast dispel second targeting mercy unless dispel can target itself.

JH has it right. You could get 6 creatures costing 1 mana each or 1 creature costing 6 or any combination adding up to 6 total mana.  {Protean Hulk}

Infinite is right, the answer is in the cipher wording. You only copy the coded card when combat damage is dealt not ability or effect damage.

https://www.wizards.com/Magic/magazine/Article.aspx?x=mtg/daily/rc/237

Here's your answer straight from mtg on a featured deck

Being the caster of spelltwine he picks how the next two spells are put on the stack then as spelltwine says "cast them if able" if mercy is first on the stack it resolves first. Dispel has no target and is unable to cast so nothing happens.

{Korlash, Heir to Blackblade} has been targeted with a fuse casted  {Turn // Burn} but it's owner pays mana to regenerate him in response to me declaring the spell. Does he live or die?

It depends how damage is applied. He gains the regeneration shield before losing abilities then you burn him. He dies and regenerates but cannot cast regenerate again so if lethal damage is applied a second time he dies for good.

If this were double strike he would take the first strike damage, die, regenerate, leave combat and tap. Normal damage would not apply for the second strike. The key here is he loses abilities for turn not effects.