The harder she clings the more she pushes him away. Eren is a kindred spirit like a bird, you can't try to force him to stay with you, you have to let him go. And who knows sometimes that bird might just come back to you.
Eren can use the 3dmg too, every military has to know how to use it (otherwise they wouldn't be able to fight titans at all), but as you saw by episode 7 (7,8, I dont remember) he "died". Mikasa is just abnormally good with this equipment, like Levi.
He died because he saw a friend dying. He got so full of rage that he lost his mind and went with all his will to slaughter that titan.... But he is not like that anymore, he matured and is learning to control his emotions, which he learned in the hard way
No he "died" because he saved Armin from being eaten. He selflessly sacrificed himself to save a friend. Mikasa fans have it written as he died because of incompetence and like to ignore what happened with Armin.
before saving Armin he had one of his legs eaten and his head was full of blood... and the way he feel and rolled through the roofs he should be full of fractures. If this didn't happen maybe many things would be different: he would manage to save Armin without dying; or he would still be running to that special titan (he could have killed it or be killed, we dont know); if he didn't die with the special titan he could be too far to save Armin, or maybe he could kill that titan from behind and rescue armin from inside its.