Oh hell no, OP. Just hell no.
1st, teenagers can be trying in a solid marriage, but a teenager in a dysfunctional divorce will wreak havoc. You may not have kids, but you were a kid. If you recognize that her behavior was out of the norm, and that was in front of company, imagine what it is like typically. My kids may pull some stuff at home, but they know that if they were to ever act that way in front of someone else the consequences would be dire.
2nd, you are only dating 6 months and you are already that involved in his kids' lives? At 6 months you should, maybe, just be meeting these kids, not already have won them over. I was dating my now husband a year before and we discussed marriage before he even met my kids. Meeting GF after GF after minimal dating just introduces more chaos into the lives of those kids.
3rd, you are not, nor will you ever be a mother figure, or anything close to those kids. They have a mother, and they are too old to develop a maternal relationship with another woman.
There is way to much drama and chaos in this man's life. He has a dysfunctional relationship with his ex and with his kids. I read your post about all the things that happened to you in a very short amount of time. You managed to extricate yourself from a chaotic and drama filled situation. why in the world would you want to put yourself back into one?
You have had a a lot of upheaval in your life. At your age (and we are close in age) do you not want some security and peace and calmness? I have 6 children (1 is an infant, 3 are teens) and 3 dogs and I do not have this level of drama and chaos in my life. Screaming, yelling, arguing, anger, these are things I have no time for.
You said your mother was your greatest supporter. Think of your mom. What would she say about this situation?