No. The battery is very well optimized and thanks to textured polycarbonate body - we have not come across any heating issue during our review.
Moto C has a decent pair of the camera on both the sides. The rear camera has 5.0 MP resolution camera with Autofocus and LED Flash. There are many other options available like beautifying software, HDR, Burst mode,etc. The front camera has 2.0MP and HDR support. The front camera also has LED flash and you can also use display screen as flash.
Yes, Moto C comes with removable battery.
No. Moto C does not support fast charging.