You can get a single entry Non imm O visa based on having a Thai child with just your name on the childs birth certificate, no need to be married or have gone through the courts.
The 60 days extention IS available if your name is on the childs birth certificate even if you're not married or haven't been through the court process. It's only for the 1 year extention that you need to go through court.
You can get the single entry Non imm O with just your name on the childs birth cert but to get the 1 year extention you need to go through the court, proving you're the father.