Yes, you will get a pair of headphones with this mobile phone in the box.
Yes, you can change the battery of Moto C. It comes with removable battery.
No. Moto C does not support fast charging.
There are many features which make Moto C and Moto C Plus - completely different phones.
01) Moto C has 1GB RAM while Moto C Plus has 2GB RAM.
02) Moto C has 2350 mAh battery while Moto C Plus has 4000 mAh battery.
03) Moto C has 5MP rear camera while Moto C Plus has 8MP rear camera.
But all these also makes Moto C Plus costlier than Moto C. If you are looking for normal usage - Moto C is a better option, considering the price. Both Moto C & Moto C Plus support OTG and also has front camera flash.