I am no expert on Eilan Donan Castle save that I know it was completely destroyed by the Navy during the time of the Old Pretender (1719) when they found some Spanish soldiers garrisoned there. It lay a complete ruin until around 1911 when an entrepreneur re-built it. (The modern chocolate box castle is a 20th century replacement. Sorry). So I am not certain what early records may exist. Probably not many. The Cox family being Presbyterian does point to Scottish ancestry.
Hannah’s remarriage to George Stegg (note spelling) was on 8.2.1871 in Belfast so that dates William’s death to prior to that.
I see two possible deaths for William Cox. 1st Oct 1870 aged 66 and 21.9.1868 aged 40. (Ages were usually juts guesses in the 1800s). Both registered in Armagh. Neither is free to view yet so you will need to pay to see them to find out which, if either, is the correct one. You can view the original certificates on-line on the GRONI website, using the “search registrations” option:
You will need to open an account and buy some credits. It costs £2.50 (sterling) to a view a certificate.
Belfast Newsletter of 16th May 1862 announced the marriage. Wm was Land Steward to Lord Gosford and Hannah was 2nd daughter to Clark Louden Esq of Dennismullen, Armagh.
One minor item. You mention the couple being married in 1862 by special licence. However there’s a civil certificate showing they married in Armagh register office (witnesses William Smith & Sarah Cox).:
There doesn’t appear to have been a church wedding.
There was a Sarah Cox death registered in Armagh on 10th June 1868, aged 36. You might want to look at it, in case it’s William’s namesake.
The Gosford estate papers are in PRONI so perhaps there may be mention of William there.
Marriage registration started in April 1845. If William’s first marriage was before that it will be much harder to trace. You need to search church records. Not all pre 1845 records have survived and of those that do, not all are on-line. Tradition was to marry in the bride’s church, so you can’t even assume it was Presbyterian. Death registration didn’t start till 1864 so you won’t find a death certificate for her either. Presumably she’s buried near Gosford Castle. I would try the graveyards near to Gosford Castle in case she has a gravestone. According to the Ulster Historical Foundation site there are 3 likely graveyards there: Mullaghbrack Church of Ireland, Markethill 1stPresbyterian and Markethill 2nd Presbyterian. (Many Presbyterians were buried in Church of Ireland graveyards so I would definitely check it).