Non-PRO Spanish to English Law/Patents Certificates, Diplomas, Licenses, CVs birth certificate

Hi there, I'm translating a birth certificate. I am confused by the following line:

Doy fe. Expido el presente en [name of city] a 5 de Octubre 1979.


My understanding of this translated into English is as follows:

I testify. I issue this document /[or]/ I testify the issuance of this document on 5 October 1979.


I am confused by a few things. One is the full stop and structure in "Doy fe." and "Expido".

Another is the fact that "el presente" usually seems followed by "certificado" or "documento" according to my research. Would it still make sense to be alone like this?

The third point is the absurd fact that the birth itself took place on 29 October 1979. How could the document possibly have been issued earlier in the same month? I believe I have misunderstood something important here.

Thank you very much for your kind assistance.