Hi Drew
Before I get on to your certificate question, can I just tell you that I took another look at Griffiths Valuation. There are two entries for a Jeremiah Hayes in Cavan. This doesn't necessarily mean there are two men of the same name. The two entries are for adjoining townlands and the same landlord. I'd guess just one Jeremiah farmed land that happened to straddled the townland border.
This is relevant in view of info below.
To order certificates from GRO in Roscommon go to
www.groireland.ie and download the relevant birth, marriage or death certificate application form. You have to print off the form as there is currently no online ordering facility.
When you fill in the form, don't worry about too many of the questions. So long as you quote the reference number you don't need to provide any more info. So, on the death certificate application form, just quote this: Jeremiah Hayes, Q1 1882 Bawnboy 3/29. Further down the page, tick the box for photocopy. Add your credit card and adddress details. Nothing else is needed.
When I went back into the LDS pilot site to find that reference for you, I found another entry that could be very interesting for you. It is also for a Jeremiah Hayes - the only other jeremiah hayes registration entry for Cavan on the LDS site, in fact. It is a marriage record dated 1857, in Cavan Registration District. This date is before civil registration of marriages started for Catholics so this marriage would have been held at the Cavan Town register office or in a Church of Ireland within the Cavan registration district.
If your family were Catholic, don't automatically assume this isn't your ancestor. Due to the penal laws, which continued even in 1857, Catholics would sometimes marry in a Church of Ireland or Register Office because that was the only way to have your marriage legally sanctioned. (Marriage in a Catholic church was valid but officially illegal. This was an important distinction when it came to inheritance.) It is also possible that this 1857 wedding was a mixed-religion marriage (and even possible that two ceremonies took place, one in each church...in which case only the non-Catholic ceremony would be accepted for civil registration purposes).
If I were in your shoes, I'd find it impossible to resist getting sight of this 1857 marriage certificate. Given that there appears to be only one Jeremiah Hayes household in the county of Cavan in the late 1850s (according to Griffiths), this could well be interesting info on the certificate. Perhaps Esther had died. Either way, the name of jeremiah's father will be on the certificate and that could be a useful nugget.
If you decide to order the certificate, download and print the marriage certificate application form, and write in "Groom Jeremiah Hayes 1857 Cavan 3/513" and send it off with the death certificate application.
Did Jeremiah come from Cork? It's impossible to say. He might have done. Or his father might have done. On the probability of numbers, it's quite likely one or other did so. But there are always exceptions....
All the best
Siobhan