The vertical piece would have to go under the chimney's counter flashing.
Installing fiberglass insulation against the stone wall should minimize or maybe eliminate condensation problem as well. One thing you have to remember is that the vent filter needs to be cleaned once in a while, as soon as it gets contaminated the air flow will stop. I don't have any experience with those vents but it appears that you should be able to lift the vent and remove its filter for cleaning... it might require removing of the chimney's counter-flashing.
Let me know if you have more questions,