So I've got two problems with my armature Please, keep in mind that I'm quite new to this whole world of rigging.

The first problem is that I've got a bone that is moving my model in a way that it shouldn't. No it's not a weight problem, Please take a look.

Here's one of bones that is working properly



As you can see, it bends model properly (or maybe not, I'll improve weights soon)



However, this little guy right here, whenever I move him.



The whole model moves along, and I seriously have no idea why?



And second issue is related to face. I also have no idea on why is this happening?



When I turn my model's face around, eyes and teeth just sort of pop out for some reason.



I've tried using Ctrl + J but it didn't help much.
A quick and easy fix of all your problems, try Rigify. It will give you complete custom controls and you may assign bones to mesh With automatic weight.

Enable Add on



Add Armature 



Adjust bones according to your character, Ctrl + A and apply Location, Rotation and scale.

Hit Generate button under Rig tab in right side bar.

Select mesh, select custom shapes (in this order) Hit Ctrl + P and select With automatic weight.

I hope it will make your life easy, Good luck :)