If Pyreswipe Hawk leaves the battlefield before its last ability resolves, control of the target artifact won't change at all.

The value of X is calculated only once, as Pyreswipe Hawk's third ability resolves.

If the cost to cast a spell is increased, decreased, or changed because of additional or alternative costs, expend counts only the mana you actually spent.

Abilities that trigger whenever you "expend N" only trigger when you reach that specific amount of mana spent on casting spells that turn. This can only happen once per turn. For example, if you've spent three mana on spells so far this turn and you control a permanent with an ability that triggers "whenever you expend 4," that ability will trigger the next time you spend at least one mana to cast a spell this turn. It won't trigger again if you spend another four mana to cast spells later in the turn.

A permanent with an ability that triggers whenever you "expend N" will see mana you spent to cast spells the turn it enters, including mana you spent before it entered. For example, if you cast Junkblade Bruiser (a creature with a mana value of 5 and an ability that triggers whenever you expend 4) by paying its full cost of five mana, you'll already have spent four mana on spells before Junkblade Bruiser enters, and therefore its ability that triggers whenever you expend 4 won't be able to trigger at all that turn.