Genealogy in Germany is a money earner for the country and you won't find many online databases, which is understandable as it brings in money for the countries archives and wages to professional genealogists. I found my ancestors by directly applying to the church archives and from then on I viewed micro fiche in my local Morman church in England at a very low cost. i was lucky that WWII hadn't destroyed 19th century property and people count = their census - so I was able to track my family in Hannover every year from 1851. Unfortunately I have not been able to find mention of any such records for your area of interest, which suggests there might not be any..
One old custom was that family names were carried over to the next generation and this might be the case for your ancestors. This was particularly true if the family were in the farming industry and the farm was handed down, so you might find that a husband of a farm heiress changed his surname to that of his bride.
The first son was named after the father's father
The second son was named after the mother's father
The third son was named after the father
The fourth son was named after the father's eldest brother
The first daughter after the mother's mother
The second daughter after the father's mother
The third daughter after the mother
The fourth daughter after the mother's eldest sister
because my ancestor followed this custom I notified the German archivist of my ancestor's birth year & that his parents could be Henry and Sophia, which turned out to be the case; additionally the less time an archivist searches for information the cheaper it is on your pocket.
I thought my ancestor was the oldest child as he was born during the first year of his parents marriage, but like you, discovered there was another handful born to the same parents out of wedlock but who lived with their maternal grandparents. From the census it seemed the parents lived in accommodation provided for single people. My understanding is that this wasn't uncommon and probably had something to do with marrying when a family house became available.
The Catholic church will have all the information whether the child was baptised in or out of wedlock. The Catholic churches also recorded "confirmations". I have some parents who lived in a Lutheren parish, had their children baptised in the Lutheren church, then seemed to have saved up to make a long journey to the catholic Hildesheim church where they had five children confirmed at the same time, the oldest child was 10, the youngest one year old. The Catholic archivist informed me that this was a usual occurrence - e.g. the Catholic church allowed this if the father worked in a Lutheran parish.
I suggest you try to find other people researching the same lines and you might get lucky by viewing the German
GEDBAS website - on the webpage simply fill in the surname, or even just fill in the place they lived which will give umpteen surnames living in the same village/town, and view the results. Here is a meta-engine search facility for you to play around with which brings up several websites :
Click on GEDBAS then Enter your surname and/or village
You could also try "Auswanderer" search results.
http://meta.genealogy.net/I see there is a micro fiche of migration of auswanderen fro your area of interest: When a person wanted to migrate they had to ask the town elders. Their names would be written up on the civic centre/town hall door for about three weeks which gave a chance for anyone else in the neighbourhood to come forward and state why the person shouldn't be allowed tomigrate, e.g. he owes me money, or (s)he is my apprentice and not allowed to leave. The information on the official documents usually gives name of the main applicant and his/her abode plus names of accompanying family members and their relationship with the applicant.
https://www.familysearch.org/search/catalog/1135289?availability=Family%20History%20LibraryI haven't done any German research for well over a decade and am not knowledgeable about modern sources. Hopefully other rootschatters will come along and provide more sources of information for you.