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.
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.
Magic: The Gathering · ™ & © Wizards of the Coast · Illus. Alex Horley-Orlandelli
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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.
