A spell you cast this way can be the target of a later spell you cast this way. However, permanent spells cast this way won’t resolve until you’re done casting spells, so the permanents they become can’t be the target of spells cast this way. For example, if you exile Twincast and Lightning Strike, you can cast Lightning Strike and then cast Twincast targeting it; but if you exile a creature card and an Aura card, you can’t cast that Aura targeting that creature.

If you cast a spell “without paying its mana cost,” you can’t pay any alternative costs. You can, however, pay additional costs. If the spell has any mandatory additional costs, those must be paid to cast it.

For chapter I of The Great Synthesis, the number of cards you draw is determined as the ability resolves. If you no longer control The Great Synthesis at that time, your maximum hand size remains unchanged.

If a spell has in its mana cost, you must choose 0 as the value of X when casting it without paying its mana cost.

Once you’ve activated Jin-Gitaxias’s last ability, it doesn’t matter what happens to the number of cards in your hand. The ability will resolve even if you have six or fewer cards in your hand by that time.

All the spells you cast due to chapter III are cast during the resolution of that ability. You can cast them in any order, and timing restrictions based on the cards’ types are ignored. Once you’re done casting spells, before any of them resolve, The Great Synthesis will be exiled and return as Jin-Gitaxias.