As I recall I had to be listed on my daughter's Dominican birth certificate, and the powers that be won't list the father without a marriage certificate. Since we were married in California, the marriage certificate had to be notarized and apostilled and officially translated and legalized in DR. So, with daughter's Dominican birth certificate could establish paternity. The rules for recording a live birth of a US Citizen abroad are paternity and residency. The US parent must have lived so many years in the US after the age of 14, or some kind of rule. The form lists all the documents that they accept as proof, however when I declared that my daughter is a US citizen some dude (in the DR) pushed a button and popped out a government record of me going back to the 1980's and used that to 'prove' that I satisfied residency requirements (which is IMHO kinda crazy that someone in the DR can do such a thing, but whatevers I guess). The guy didn't care about any of the 'residency proof' documents I brought with me.
When i submitted the declaration, I had to swear that I would be financially responsible for my daughter until she is a legal adult. So that's the answer to the question about child support. Why would a parent not want to support their child, anyway.
The legwork and documentation was much more intense in the DR, because they do things differently. For example, have to make 3 trips downtown for a simple documentation thing, and they want to keep your original notarized and apostilled documents which, in my opinion, makes it easier for crooks to steal identities. In the US they want to see original documents, and maybe make photocopies, but they don't keep original documents. IF you show up at a government office with a photocopy of a birth certificate, they aren't going to accept it, right? Even the school here wants an original notarized birth certificate for my daughter. What security measured do they have on those documents? Probably employee identity theft waiting to happen. Anyhow...
About the OP, the "three date rule" has been OK with me, but women who don't put out on the first or second date are probably not going to get along with my shenanigans anyways. So, I don't usually bother with third dates if nothing's happening. I'm just saving the woman the trouble of thinking they might want to like me.