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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.

When Herald of Leshrac's cumulative upkeep ability resolves, if its age counters outnumber the number of lands on the battlefield that you don't control, you can't pay its cumulative upkeep cost and must sacrifice it.

Herald of Leshrac's leaves-the-battlefield ability affects all lands you control but don't own, not just the ones you gained control of with Herald of Leshrac. For example, if you had gained control of an opponent's land with Annex, its owner will regain control of that land. Annex will remain attached to it, but its effect will be overridden.

You must choose a different land you don't control for each age counter on Herald of Leshrac. Otherwise, you'd try to gain control of a land you *do* control midway through paying the cost and need to back up.

The cumulative upkeep trigger doesn't target what you gain control of, and you don't choose lands until the ability resolves. Your opponents can not tap lands in response to the choice, but they could respond to the ability by tapping any number of lands before you've made any choices.