Clearly, the quote above is from STIII when Kirk fights Klingon Commander Kruge to the death on the Genesis Planet while everything lights up in flames around them. Kirk says it in the final stage of the fight before sending Kruge into the abyss.
Yep. While Kirk is certainly shown to be willing to extend mercy to an enemy he's not willing to do so at the cost of his own life when that enemy refuses said mercy.
On an amusing personal note, "Kirk on a volcano boot stompin' a Klingon" has long been a shorthand between my wife and I when we are indicating that we are totally sick of someone's shit but might be overheard.