Establishing Paternity: Man Signed My Daughter's Birth Certificate

My question involves paternity law for the State of Oregon. I was engaged to the mother of my daughter when we found out she was pregnant. Four months into the pregnancy I discover that she is cheating on me, and she ended up leaving me for the other guy. This occurred when we lived in Montana where my daughter was born, but she has lived in Oregon for the last 9 years of her life. The mother lied about letting me be at all of the doctor visits, and every aspect of the pregnancy, and didn't even let me know when my daughter was born. I found out the night she was born and was denied access to the hospital. After about 3 weeks of letting me visit my daughter she disappeared and I didn't get to see her again until I was reunited with her a year ago. The man who is the "legal" father is very controlling and an alcoholic, and he and her mother have divorced so now the mother is out of the picture completely, so I want to remove my daughter from that environment. However, he won sole conservator over my daughter and her three half-siblings due to the fact that the mother had started using drugs heavily, overdosed and was dead for two and a half minutes before being revived, and still continues to use while rarely even making an effort to see the kids. My daughter is now 11 years old and I want to take over her upbringing.

So, what I would like to know is what, if anything, can I do to prove paternity and take custody away from him? Once I do, will the birth certificate then be modified to replace his name with mine, change my daughter's last name, an grant me custody without having to sue him for custody?

Also, will this fall under Oregon Law or Montana Law?

Thank you in advance to anyone who is willing to help point me in the right direction, God bless!

Establishing Paternity: Man Signed My Daughter's Birth Certificate