Do Cable Connectors/Terminal Blocks impact power requirements?
Terminal blocks need to be rated for the current they will handle (otherwise they can melt).  For power levels you'd see in an ecigarette, pretty much any will work.

Your bigger problem will be using AA batteries.  Most Alkaline AA batteries will not work very well at high currents.  Particularly cheap ones like dollar store AAs - they are meant for remote controls and other things that draw very little power.  That's why ecigarettes typically use lithium ion batteries.  At high currents, the battery's voltage will drop very quickly.  

High quality rechargable batteries handle this a little better - I had a 4xAA mod using PowerEx rechargeable batteries that worked fairly well, but not nearly as well as a single 14500 Lion cell.