That space and sepperation from base to the tank bottom collar is due to installation of the wrong head. The T3S hip of its base, the section that mates with the flange at the bottom of the coil head is narrower than Protank and Evod coil head, it is very hard to distinguish and a lot of vendors sell both, a lot of times selling the wrong head, or they think that all Kanger coil heads are the same. A proper T3S coil head when mounted onto the base the flange at the bottom should be flush and even with the top of the T3S base. If the flange overhangs the top of the base by any margin, the base and head when assembled will not seat fully into the T3S tank bottom collar giving you what is in the picture you posted.
On to the issue of the coil not firing, crxess answered it properly, get you PV pin pulled back up, or method I use, gently and carefully wiggle the T3S connection pin out just a bit, seat the tank and let that seat the T3S pin to where it needs to be. Air draw, again crxess is right, a 510 to Ego adapter with a hollow center pin works best, another thing is you can increase the size of one of the airholes on the T3S base, I enlarge at least 1 to 1/16th inch, if that doesn't help, then I'd drill a hole into the top of your 510 to Ego adapter, above the Ego threads through the Female 510 threading section.