There is more than one efest 14500, what color is the plastic wrapper?
Now!:
The first step in sorting this out is to REMOVE the battery from the PV.
If your mini Nemesis is built like a real Nemesis the top of the PV is removable. This should be the only part of the PV that you need to work with.
SO; remove the top cap and install the eGo adapter. Grab your little continuity tester (you do have one I assume), and see if you have an open or a short. You should have an open circuit still. If you have a short at this point the adapter is bad. If the adapter is OK then install your T3. With the T3 installed you should read the resistance of the coil. If you read a short, the T3 is bad. (or the connection between the adapter & the T3 is bad)
It's really easy to test all this with the top cap removed so that you have easy access to the positive post of the PV. (and the battery can't blow up while you're sorting the problem because it isn't there)