I had to send one letter to some lawyer company, and they answered me that .. so what's exactly the adoption at District Office ?

" If the mother lists you as the father on the birth certificate then you can register at the District Office as the father and apply for adoption. Obtaining custody would have to be done through the Family Court and would depend on both the mother and yourself. In general, unless the mother has been proved unfit the courts are unlikely to grant you sole custody. You could however apply for shared custody once you are declared the father "

It is not an adoption.The lawfirm is pointing out that with legitimisation of the child you become the legal father, but not necesarrily have custody of the child. It are two different matters.

I think that if you are married, you will have dual custody automatically. But when not married, you might need to get formal parental powers from the court.

Normally you don't need to get formal parental powers if the relationship with your girlfriend is good, it is more important when things run sour and there is a dispute about who will get the child how many days.

If you want an extension of stay from immigration based on your child, you will need to have formal custody over the child and have to go through court.

So, I'm sorry.

If I go to District Office with my girlfriend, I'll be a Legal Father without custody right ??

That seems correct, but I'm not 100% sure on this. It is an interpretation of the law I'm not certain on how Thai courts will act upon.

You do know that for legitimisation at the amphur the child must also agree and for that needs to be at least around 7 years? (There is no fixed age for this, but most amphurs seem to accept from age 7).

As said, for immigration purposes going through court is advisable. They don't accept a legitimisation at the amphur and the custody issue might be the reason for that.