Statutory recording of deaths only started in Ireland in 1864. So no death certificates before that year. Presbyterians generally didn't keep burial records, so you are very unlikely to find any death records for family who died in the late 1700s or early 1800s, unless they are named on a gravestone or memnti0oned in a newspaper (wealthy folk only).
You are right to assume that if a name was re-used it often indicated the first child had died. Families liked to keep traditional names alive, literally. There are exceptions, sometimes where a family was following the Irish & Scottish naming pattern strictly you got 2 children of the same name (because say both grandparents shared the same forename) but in my experience that’s not terribly common. Where you find more than one with the same name, that the firstborn had died is usually the explanation. But as I say you will struggle to confirm that from church or other records, since such deaths were rarely recorded.
Boardmills 1st Presbyterian church has baptism and marriage records from 1782 onwards. There’s a gap in the marriage records from 1814 to 1823 but the baptisms are complete. There are no burial records.
I don’t think the baptism & marriage records are on-line anywhere but there is a copy in PRONI (the public record office) in Belfast. A personal visit is required to access them. Access to the records there is free. This link explains what records exist, parish by parish:
If you are unable to go yourself, you could employ a researcher. Researchers in the PRONI area: http://sgni.net
The 2 children baptised on the same date might be twins but they might just be siblings in a double baptism. No easy way of knowing with Presbyterians . (They didn’t necessarily hurry to baptise. I have seen 6 children baptised together whose ages spanned 8 or 9 years).
The RosDavies site records 2 Brice McKee marriages and some children:
Looking at that, it seems to me that there were 2 separate couples both living near each other in Killaney parish. One was Brice McKee married to Mary McKee and another Brice was married to Mary Tannihill. That may explain some of the name duplication but it still looks as though some children died young and names were re-used. There was another Brice in what looks like a later generation.