If Arthur Petrelli has Adam Monroe's power to heal, then how come he died when Sylar shot him in the front of his head?

Easy. The Hatian was blocking Arthur's abilities, preventing him from healing.

Okay say Claire was shot/stabbed we all know she has the power to heal herself but the only way she can heal herself is if the bullet/knife were to be removed.
That's just how the power works, there can't be anything as such in your body.