Hi Anthony,
I can't remember the percentage of Azoreans who are now Roman Catholic. It's high. Back in the day, it was probably something like 98% were Catholic. Maybe someone has those statistics.
At the time period you are speaking about, there are no birth records. Birth records didn't start until 1911. Before that, we use the Catholic Church baptisms as our source for birth records. So a baby would be born at home and within a few days to a week (*typically* - there are always exceptions), the baby would be taken to the local church to be baptized. Since your ancestors had already left the Azores, they would have to baptize their baby when they arrived in Australia. You need to check the Australian Catholic Church records near the port where your ancestors arrived.
The Azores islands do have some regional differences, just like many countries have regional differences.
Farming, fishing were very common and many still do farm or fish now.
Manila? North South West? Is that the Philippines? I thought you were talking Australia.
For arrival documents, you'd have to check your country (Philippines or Australia or whatever you are talking about) and ask about their arrival records or immigration records. A local genealogy society or your local Family History Center should be able to help you. If you don't know what a Family History Center is, look up Church of Jesus Christ of Latter Day Saints (Mormons) and call them and ask where the nearest Family History Center is. No, you don't have to be Mormon/LDS to use it. It is open to the public like any other library, however many have limited hours since they are run by volunteers.