Among abilities, only activated and triggered abilities use the stack. Activated abilities are written in the form "[Cost]: [Effect]." Triggered abilities use the word "when," "whenever," or "at." They're often written as "[Trigger condition], [effect]." Some keyword abilities are activated abilities and will have colons in their reminder texts, and some keyword abilities are triggered abilities and will have "when," "whenever," or "at" in their reminder texts.
As Whirlwind Denial resolves, first the opponent whose turn it is (or, if it's your turn, the next opponent in turn order) chooses which spells and/or abilities to pay for, then pays that amount. Then each other opponent in turn order does the same. Then all spells and abilities that weren't paid for are countered at the same time.