I can really only help you with one question. You ask if the family were Ulster–Scots. The name Paisley is reasonably common in Scotland, the family were Presbyterian and they lived in Co Antrim. Those 3 factors together make it 99% certain the family were Ulster-Scots.
Presbyterian is by far the most common denomination in Co. Antrim (even after 200,000 of them emigrated) and effectively all are descended from Scots settlers who came to Ireland in the 1600s.
The births and the marriage are all well before the start of statutory registration in Ireland and so you are largely dependant on church records. Not all churches have records that go back to the early 1800s and not all are on-line. Most surviving records have been copied and are held in PRONI, and Belfast.
If you have no other leads, given the links with Co Antrim, I would focus on where Paisley families were to be found there in the 1800s. Griffiths Valuation (1860s) shows there to be just 9 households spread across 5 parishes. If you have no other leads you could try going through those church records to see if you can find your family.
Bear in mind that Presbyterians in Ireland don’t use the parish system, and which church you attend is simply a matter of preference. It’s not always the nearest one to where you live. And places like Carrickfergus have several Presbyterian churches. Belfast is a bit of a nightmare as it has over 50. However not all were in existence in the early 1800s. Rural areas are a bit easier.