1. One strong (expensive) servo. Well mounted gun, so center of mass is on rotation axis. For 500g-1000g gun should work, but it’s more a guess, than calculation.
2. I very much doubt that any kind of ranged heat detection will work (other than professional thermal camera). In link it says it "can detect a candle flame at a range 2 meters (6ft) - unaffected by ambient light". Candle flame has tamp. is over 1600 Kelvin (1327 C). Human has 309 Kelvin (36 C). Background temp. in room is 293 Kelvin (20 C). Candle flame is small and hot on cold background. Human is quite big (compared to candle flame) and just little warmer than background.
If someone DID use IR to detect heat, please list exact sensor, link to documentation, price and effects.
I would suggest to mount IR transmitter on target, and 3-4 IR detectors on gun, separated by tiny walls.
3. If sensor 1 detects IR signal, rotate left fast, If sensor 2, rotate left slowly, If 2-3 sensors detect IR at same time, but 1 and 4 not, then shoot.
4. You can use one microcontroller for detection and controlling servo.
I suggest you first build sensors, write program and test it. Leave gun, trigger, power, servos for now. These problems are easy solvable. Sensors will be the most difficult and time consuming part.
You can program video processing by yourself on PC. I advise not to try use video processing on microcontroller. It's possible, but very very difficult.
If you want to use camera (IMHO best but most complicated sensor), connect whole gun to PC, and use some ready libraries (e.g. robotrealm).