Thank you for the good information from the documents that you have.
It is useful to identify the author of the information on birth and death certificates. Who is the informant named on the birth certificates and death certificates that you have?
"Previous issue" is used on birth certificates, and records earlier births to the same couple. On the birth certificate 1867, you have listed this as "children".
You do not need a certificate for every BDM event in the lives of the people you are researching.
It is useful to target particular records, keeping in mind the event -
births........earliest evidence you have of the person.
marriage...people are giving information about themselves ie. self-reporting.
death....end of life record....information about the persons origins, and what happened since they were born. This is a story as recorded by the informant, and as known to that person.
during the 19th century in Australia this will vary from state to state, and change over time as governments wanted to record more information about people. But some did not record much at all eg South Australia and Tasmania.
Mary WALSH died in 1888 as a widow. The person who knew her longest would be her husband James, and he is not around to be the informant.
James WALSH died in 1882. The person who knew him longest would be his wife Mary. Mary might be named as informant, but if she is not, then you might assume that she is the source of information on his death record. You do not need to get his death certificate, but I am suggesting how you might consider which certificates to get.
The birth certificate, 1867, is useful to compare with the death certificate, 1888 to account for children.
Death certificate for Mary WALSH, 1888...it is interesting to see that it has family name ...O'CONNOR...for her mother, not no given name. You might think that knowledge of Mary's origins is passed on in conversation by Mary during her lifetime, and her mother's given name would be known?
Does that suggest that Mary has received knowledge of her mother, but no personal memory?.
Was Mary brought up by father Jeremiah and a stepmother? Just ideas to ponder.
Use PROV (Public Records of Australia) to find a probate records for Mary WALSH, died Timor West 1888.
https://prov.vic.gov.au/explore-collection/explore-topic/wills-and-probatesMary owned land when she died.
I am not seeing a probate file for James WALSH who died 1882? If James died with no estate, you might wonder how and when Mary acquired property.
Ambrose is a less common name and might repeat the name of a brother to his parents James WALSH ....Mary MURPHY.
Can you give the names of informants for the birth record, 1867, and the death record 1888.