Re: Stockholms stadsarkiv assistance
To look for people in the older church records I often use the indexes called Kyrkoböcker at Stockholms stadsarkiv.
I have downloaded all 237 OCRed PDF files so I can do a search in all of Stockholm at one go.
The Barnbördshus images come from SVAR (http://www.svar.ra.se):
Allmänna barnbördshuset, Förlossningsjournaler, SE/SSA/1932/F/F 1/F 1 a/16 (1804-1805), bildid: C0054135_00095
Yes, I think that Hedvig was the birth mother.
Fathers are always hard to find when they don't want to be found. It is possible but not likely, I think, that there is more information in records that have not been scanned yet. The number 1473 on Gustava's orphanage record refers to a register of "Spädbarn" (Infants) and Stockholms stadsarkiv can help you with that.