Ralph,
The majority of migrants from Ireland to north America in the 1700s were Presbyterian. (Native Irish didn’t start leaving in significant numbers till the 1800s). So leaving in the 1700s, being Presbyterian, having a Scottish surname and living in Donegal (a county that many Scots settled in) all point strongly to Scots ancestry. So you should expect that they arrived in the 1600s either as part of the Plantation or possibly later. (There was a large influx in the 1690s due to famine in Scotland). Your ancestors would generally be known as Scots-Irish in north America and Ulster-Scots here in Ireland.
Looking at the 1630 Muster Rolls, there were 4 Shaw households in Donegal. All in the same general area. Headed by Anthony, Gilbert, James and John Shawe (younger) respectively. They were all on the Earl of Annandall’s estate in the Baronies of Boylagh & Banagh. The Earl had 10,000 acres there. I have attached a link to a map of the county which shows where those 2 baronies are:
https://cotyroneireland.com/maps/baronies.html
The numbers on the maps refer to the individual parishes within each barony. This link explains what church records exist, parish by parish:
https://www.nidirect.gov.uk/publications/proni-guide-church-records
The Muster Rolls describe all the Shaws in Annandall as having “no arms.” The rolls describe the weapons that the men had to fight with. Normal categories included a sword, snaphance (a primitive musket) and a pike. If someone had “no arms” ie no weapons (and many were like that) it suggests they weren’t very wealthy. Probably labourers or very small farmers. To be in the muster Rolls indicates you were of fighting age so perhaps aged 16 to 45.
You ask about the Genealogical Office in Dublin. There isn’t anywhere that carries that precise title today. There are various repositories such as the National Archives & National Library where records are held but in general you have to go there in person, or pay a researcher to go there, in order to access them. Here’s a link to some professional genealogists based around Dublin:
https://accreditedgenealogists.ie/members/
The wills records you ask about are on-line on this site:
http://census.nationalarchives.ie/search/dm/home.jsp
Until 1858 all probate processing in Ireland was dealt with by the Church of Ireland (regardless of your denomination or whether you were a churchgoer at all). The wills themselves were stored in the Public Record Office in Dublin and destroyed in the 1922 fire. All that remains, for most files, are these indexes which were obviously stored somewhere else. For someone residing in Philadelphia to have a probate file in Ireland indicates that they still had property in Ireland. Probate in this case was granted in 1787 but the actual death may have been a year or two before that. Most people who left Ireland permanently weren’t very well off and took what little they had with them. Economic betterment was the main reason for leaving, so few left any property behind. This Archibald Shaw might have been a merchant or something like that. But he would definitely have been better off than the average emigrant in the 1700s. You could try searching newspapers for an obituary or a report of his death. Unlikely to be an obituary in Ireland but possibly in Philadelphia?