dark brotherhood quest - i have to buy some things (employ the services of), i bought them all, he still has an arrow over his head
dark brotherhood quest - been tasked to kill a guy, i killed him and he's lying dead with an arrow over his head
random quest - collect a forsworn heart. i have 3, took one and the dead body has an arrow over his head


so uhh .. whst do?

The first one.. the arrow goes and the quest completes when you select the only remaining option so once again ask to buy stuff then select that option and the quest should complete.
I can't remember exactly what that option says but it's along the lines of "there's nothing left to buy, I'm done".

Second one: Not sure, was you just supposed to kill him or do something else first like talk to him or kill him a certain way?

Third: I've heard other have this problem, have you tried killing a different briarheart and looting his heart?