After Necklift Temple Artery Burst. Now Have Feeling That Blood is Dripping Under Skin

You need to address your concern with your surgeon. Normally internal bleeding does not give sensations as you describe. If it feels that blood is running near your ear it sounds that the nerves are "playing tricks" on you which can happen after any surgery (dysesthesia), unless you see of course blood on the skin and then you need to call your doctor immediately.. Often that resolves by itself but it can take months.

This is probably a paresthesia-sensory feelings  due to irritation of the nerve. It will probably go away, but you should be followed by your doctor.

Occasionally when the artery is injured, accompanying nerves can be injured as well giving you "ghost" pain or referred pain or dysesthesias. It is difficult to diagnose this over the internet and you should be seen for evaluation. Oftentimes these aberrant sensations do resolve.