If multiple players are allowed to cast the same card, the player with priority at any given time determines who can cast it. The player whose turn it is has priority as each step and phase begins, and they receive priority after each spell and ability resolves. This means immediately after chapter III resolves you’ll have priority to cast spells before any other player does.

Urabrask will count any instant or sorcery spells you’ve cast during the turn, whether those spells resolved, didn’t resolve, were countered, or left the stack some other way. Urabrask won’t count copies of instant and sorcery spells that were created on the stack and not cast.

After chapter III of The Great Work resolves, casting spells from graveyards follows the normal rules for casting those cards. You must pay their costs, and you must follow all applicable timing rules. For example, to cast a sorcery spell from a graveyard, you can do so by paying its mana cost only during your main phase while the stack is empty.

Urabrask’s second ability isn’t a mana ability, even though it causes you to add mana. It uses the stack and can be responded to.