Officially civil marriage certs just recorded father's name and occupation, not status i.e.. alive or not, however some ministers/priests/registrars did sometimes note this - as you can see for John & Lizzie's marriage it's not noted either way. Sometimes on marriage records when a father is deceased the officiant might just record just the name and omit the occupation.
Civil registration of births started here in 1864, and Lizzie was likely born 1860* or earlier so too early for a birth record, so a search for a baptism would be required. Protestant baptism records often do not note mother's maiden surname. Given the timeframe and where the marriage took place (Methodist Church) . I would suspect that Lizzie may have been baptised in an established Church - i.e. Church of Ireland.. the difficulty is establishing where this took place...
Civil registration of non-Catholic marriages & registry office marriages (presume applies to Thomas & Elizabeth) started here in 1845 should be available on www.irishgenealogy.ie , I did have a search on here and on an Index of marriages, but found no likely match - they may have married before 1845, or possibly the information we have re Lizzie's mother is not accurate - unfortunately not all protestant church records are online and not all survive..
Added - If Thomas died after 1864 there should be a Civil Record - IrishGenealogy from 1871 onward or GRONI Online for 1864 - 1870 (the latter is a pay-website). If Thomas was actually a manufacturer then he may have had a will, I had a look on PRONI but did not see a good match
** Lizzie's age is not noted on the marriage but would seem to be 'full age' , so 21 or over.