Elizabeth,
Hugh?s marriage and death, and the children?s births are all before the start of statutory birth registration in Ireland. You?ll need to rely on church records to trace and trace the births and the marriage. You don?t say what denomination the family were.
I don?t think any of the church records for Saintfield parish are on-line anywhere, as far as I am aware. Copies of the surviving records are held in PRONI (the public record office) in Belfast. The Church of Ireland records start in 1724, 2nd Boardmills Presbyterian in 1846, 1st Saintfield Presbyterian in 1854 and 2nd Saintfield Presbyterian in 1831. The RC records start in 1837. None of these records are on-line at PRONI and a personal visit is required to search them.
Death registration didn?t start till 1864 so you won?t get a death certificate for Hugh. Only the Church of Ireland kept burial records, so if he is not of that denomination, then you would have to rely on finding a gravestone for him.
There?s quite a few Baillie families in Saintfield parish area on Ros Davies site. These might prove to be related if you can trace yours to a specific townland.
http://freepages.genealogy.rootsweb.ancestry.com/%7Erosdavies/
Elwyn