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I have no record of her birth or christening but believe her Father was Henry Coates and that she had a Sister Jane Ellen.

To find a record of her birth you would need to consult the Ballyphilip microfilms on PRONI. You may find Henry's burial (the father clearly stated on her 1887 marriage) on RootsIreland https://rootsireland.ie/down/online-sources.php

Not sure the below is the same Henry as your Henry is described as a labourer on the 1888 marriage
Henry Coates was the immediate lessor of many of the houses in Portaferry on Griffiths Valuation printed 26 Sep 1863 (in several different Townlands).Henry Coates appears in multiple newspaper articles on British Newspaper Archve/Findmypast. He owned a Hotel & shop was a spirit merchant in Portaferry. He had been carrying on the business for >25 years when let the Hotel/shop 1862.

Belfast News-Letter 1 March 1875
COATES - Feb 25, at the residence of his son-in-law John Cole, Diamond, Clones, Henry Coates, Clontivern House, Co. Fermanagh, formerly of Portaferry, in the 83rd year of his age. [that News-Letter article will be viewable on standard Ancestry] His wife Anne died at John Cole's 8 June 1872 and was also of Clontivern House, formerly of Portaferry.

Belfast News-Letter 12 November 1873 has an advert from Henry of Clontivern selling his 23 houses in Portaferry currently leased out.
Feb 17, 1857 by Rev. John Orr, Portaferry, Mr John Cole of Irish-hill Dundonald to Jane Coates, eldest daughter of Henry Coates, Portaferry. Belfast Mercury 20 Feb 1857 & Belfast Weekly News 21 Feb 1857.

https://www.irishgenealogy.ie/view/?record_id=8f6abc883a-1986866 John Cole's marriage actually at Glastry Presbyterian, Innishargy Pa.

Mary Lennon also has a public & private LennonWylie Facebook page, she is a single OAP who has done all the typing and scanning herself as a hobby / something to do. Her son assists her with the website IT.
I see she has rectified and responded to your reported error on the Guestbook.
https://www.facebook.com/groups/lennonwyliepublic/announcements?locale=en_GB

Note this question about https://www.lennonwylie.co.uk/ is specific to Northern Ireland so perhaps should be in the Irish Board.

The 'definititive listing' of what existed for where was:
Census Schedules and Listings, 1801-1831: An Introduction and Guide
First published in 2004. Re-published in December 2012 by Department of History, University of Essex.
https://www1.essex.ac.uk/history/documents/research/RT2_wall_2012.pdf

The 1929 Local Government Act abolished the Poor Law system

in England and Wales.

N.I Board of Guardian Records run later than that up to 1948 and the creation of the NHS.
https://www.nidirect.gov.uk/sites/default/files/publications/Your-family-tree-series-information-leaflet-13-poor-law-records.pdf

All Workhouses had an infirmary and that was the only public facility for births to married or unmarried mothers, otherwise it was home birth assisted by licensed or previously unlicensed midwife/relative/neighbour, a private maternity clinic paid for by the parents, or a charitable institution.
Single mothers didn't so much as choose to go to the Workhouse to have their children, it was for many their only option. For married mothers the few nurses there had some training and there was a degree of hygene.
The Midwives Act (Ireland) was not passed until 1918. It controlled the training and registration of midwives. It also forbade unregistered midwives to practice and outlawed the use of ‘handy-women’.
Midwifery 1806 – 2006 https://pmc.ncbi.nlm.nih.gov/articles/PMC1891762/

Think you are too early.
Ancestry has a collection UK, Registers of Births, Marriages and Deaths at Sea, 1844-1890 that a relation's birth & death appear in 1877 en-route to Queensland. He also appears in Australia, Death Index, 1787-1985.
For later maritime deaths of crew there has sometimes been an approximate location for the death but no record of when or where the burial at sea was conducted.
https://discovery.nationalarchives.gov.uk/details/r/C3196
https://discovery.nationalarchives.gov.uk/details/r/C3197
For Richard Elphick there is no death record on Ancestry or Findmypast.