My first question would be, how old is the chimney?
Reason I ask this is that boring holes in a brick chimney could weaken the structure of the chimney itself.
I have to agree with the comment of ghost about the cable being un-supported on a vertical run, you can't merely have a cable hanging in mid-air.
#1) Structural concerns are for real. But with out a permit,a warranted consideration? (from an official stand point)
#2) Aren't all 'snaked' in old work cables 'unsupported'?
Balloon construction etc.
#3) My main concern, Personally,Is that flue liner double insulated?
3A) For the reason I can not assure 4" of clearance the whole way.