Does the Powerjack give you health when you kill enemies with the flamethrower

To this day, it appears to still not be working from around early November 2011. Even looking at the achievement through Steam, as suggested here, didn't show the numbers. It is, however, possible to see the progress tracked through the game's HUD.

While in game: menu -> view achievements, and there will be a box you can tick that says show in HUD next to the pyromancer achievement.

I can personally say I've gotten Pyromancer in the past year, past November 2011, and it does show mine at 1,000,000 while in game.

As of the Gun Mettle Update, yes.

According to the wiki page on the update, the Spy will now receive 20% damage resistance while invisible regardless of the watch used. He will also receive reduced time on debuffs such as Jarate or Mad Milk while invisible.

As for the Dead Ringer, two changes have been made to it in regards to damage:

Changed: Initial attack that triggers feign death has its damage reduced by 50%

Changed: Damage resistance on triggering feign death scales over time. 65% to 20% over 3 seconds

Another change added in the Tough Break Update means that the damage resistance for the Dead Ringer has been further increased.

Changed: Initial damage reduction when feigning death increased to 75% from 50%