Can add a bit to what akanex2 has already posted.
"Moneydig Churchyard - current Presbyterian Church (congregation only began 1835 - Presbyterians previously attended church in Garvagh or at Boveedy in Tamlaght O'Crilly parish)"
People from Aghadowey congregation also joined Moneydig when it was formed.
"Magistrate & Clergyman cert'g correctness of inform: Robt Ogilvy of Ploodbank?, Garvagh and Rev G Smith of Glebe Stilerea ?, Garvagh"
Robert Ogilby lived at Woodbank which is just outside Garvagh
Rev. George Smith was Church of Ireland minister but not sure what 'Glebe Stilerea' should be- perhaps Glebe House? (rectory)
"Householder cert Character: Anty McAlister, Meteor, Garvagh and Robert Moore of Garvagh "
Lots of McAllisters (various spellings) about Garvagh but never heard of anywhere called 'Meteor.' Wonder if this might be Anthony McAllister of Mettican? (not far from Woodbank House) Anthony died 1863 and was the father of James McAlister (c1830-25 Oct.1860 Hong Kong), surgeon H.M. 3rd B.N.I. (perhaps the James McAlister you mentioned)
My first thought is that McIlhinney is a Donegal name but the details you have are certainly Garvagh, Co. Londonderry which isn't near the Donegal border.
"Coolbawn, County Drumgarra, Ireland" wonder if this could be the townland of Cuilbane (also known as Coolbane) outside Garvagh towards Kilrea? It's in Desertoghill Parish and belonged to the Worshipful Company of Ironmongers. In 1863 A. McAlister (possibly Alexander McAlister of Tirkeeran) was in partnership with John Witherow there. Other tenants were John (now Widow) McGuigan, Bernard McNicholl, Widow Rose McNicholl, J. Craig and Samuel Campbell, John Craig, Widow Martha Shannon, Alexander McDonald, Anthony Toghill, Charles Boylan, William Mulholland, John Boylan. If this is the correct place (and I think it could be) Martha's family were either away before 1863 or perhaps she was just working and living there at the time of her marriage.