Mesh warps when I lift the shoulder up

Yeah, the origin (green circle thing called Right arm) is where your arm will rotate around. Changing that position should help determine how and where it rotates from. It also looks like your elbow tag is outside of your arm - I would move that to where the elbow actually is. And finally I would play with hinge vs. weld attach styles when you have your arm selected in the upper right corner - one might work better for your character than the other. Adding the shoulder tag as suggested should help with Limb IK calculations as well.