Example: When core set Gandalf leaves play at the end of the round can I trigger Horn of Gondor? (In agot there is no response window but is there in lotr?)

Yeah, responses don't need windows. Actions need windows, but responses interrupt the normal flow of the game. You can use them whenever their triggering event occurs. So yes, you can use Horn of Gondor when Gandalf leaves play at the end of the round (or any other time).