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If an ability would produce one or more mana of an undefined type, it produces no mana instead.

Some abilities produce mana based on the type of mana another permanent or permanents “could produce.” The type of mana a permanent could produce at any time includes any type of mana that an ability of that permanent would produce if the ability were to resolve at that time, taking into account any applicable replacement effects in any possible order. Ignore whether any costs of the ability could or could not be paid. If that permanent wouldn’t produce any mana under these conditions, or no type of mana can be defined this way, there’s no type of mana it could produce.

To “tap [a permanent] for mana” is to activate a mana ability of that permanent that includes the {T} symbol in its activation cost. See rule 605, “Mana Abilities.”

An ability that triggers whenever a permanent “is tapped for mana” or is tapped for mana of a specified type triggers whenever such a mana ability resolves and produces mana or the specified type of mana.

A replacement effect that applies if a permanent “is tapped for mana” or tapped for mana of a specific type and/or amount modifies the mana production event while such an ability is resolving and producing mana or the specified type and/or amount of mana.

If an object’s enters-the-battlefield triggered ability or replacement effect refers to X, and the spell that became that object as it resolved had a value of X chosen for any of its costs, the value of X for that ability is the same as the value of X for that spell, although the value of X for that permanent is 0. This is an exception to rule 107.3i.

The tap symbol is {T}. The tap symbol in an activation cost means “Tap this permanent.” A permanent that’s already tapped can’t be tapped again to pay the cost. A creature’s activated ability with the tap symbol in its activation cost can’t be activated unless the creature has been under its controller’s control continuously since their most recent turn began. See rule 302.6.

The untap symbol is {Q}. The untap symbol in an activation cost means “Untap this permanent.” A permanent that’s already untapped can’t be untapped again to pay the cost. A creature’s activated ability with the untap symbol in its activation cost can’t be activated unless the creature has been under its controller’s control continuously since their most recent turn began. See rule 302.6.

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.