Most 15 year old girls in the world are not easy.
Do not underestimate the psychological and familial damage you have caused. I also suggest you pray that she, or someone she tells, does not call child protective services on you.
Grabbing by the neck is never o.k.
Stop being such a snowflake. Daughter will get through this just fine. In fact, it might be a good reminder to her that it's wrong to run her mouth aggressively and disrespectfully at her parents. Keep pampering her like a snowflake and she'll only get bolder in her defiance. People aren't nearly as fragile as you pretend. Whatever the culpability on part of the OP, the fact is that GOD allowed this to happen to the girl, and He only allows things for people that are ultimately for their own good. He allows people to suffer and to experience traumatic things. Even if I were to, say, unjustly and sinfully kill someone, the fact that I would be morally culpable of it does not change the fact that God (using my sin) willed for that person to die on that day. Some kids are so arrogant and defiant that they might even benefit from a public caning. You might be on the path towards being one of those who would start questioning "Why would a Merciful God allow this?" after a tragedy and ultimately lose your faith.
Even if the OP committed some kind of sin, the daughter should be reminded that she shares in the culpability for that sin by having provoked him.
If I find any fault with OP, it's in setting the whole thing in motion with by going after ɧoɱosɛҳųαƖs as "freaks". Even the greatest sinners deserve our compassion and prayers. One can easily reinforce how grave a thing that is without using that kind of language. We don't know how that person got that way, and there but for the grace of God go we.