What are the requirements for first-time application for a Non-Imm O visa in Thailand based on having a Thai child?

If you will marry the child's mother I'd say seeing as you're on a 60 day free entry (visa exempt) stamp now;

1 - take the one 30 day extension you're allowed,

then if you need more time

2 - take the one 60 day visit thai family extension.

That's close to 5 months to get married, get the funds in order, and either apply for the Non-O visa in country then OR go to a thai consulate in a nearby country to get one and then go for the year extension based on marriage to a thai

When a thai child is born here to an unwed mother (makes no difference what nationality the father is) the thai mother gets 100% of the parental rights.

Doesn't matter that your name is on the birth certificate, that does not convey parental rights, so you can't use "parent of a thai child" to get either the 90 day Non-O visa or the year extension after that at the immigration office inside thailand

You either marry the mother (when even after the child is born you get automatic parental rights) OR you go thru family court to get parental rights by court decree

Now with that being said, these would be your options

You COULD get one 60 day "visit thai family" extension at the immigration office where you live with just your name on the birth certificate, the mothers i/d, house book etc, and 1900baht.

AND

You could go to a thai consulate in a nearby country and get a 90 day Non-O visa based on being the father of a thai child by using the birth certificate, the mothers i/d and house book and showing 400K baht in a thai bank account in your name only (or the equivalent in an account in your country)

before we get in to the other questions you asked we would have to know if you are married to the thai mother or not.