Great finds… one of the Co. Dublin parishes that was not included in either the RootsIreland or IrishGenealogy sources
Registration of Catholic Marriages commenced in 1864, and before that registration only applied to non-Catholic and registry office marriages. The Parish Register entry would be the only official record of the event. Sometimes the witnesses to the marriage can be family and provide clues, as can the godparents for baptisms, which can provide possible leads to family. One of the godparents to Jane in 1851 is a Cooke, although the firstname is abbreviated and unclear. IrishGenealogy have it transcribed as Thomas - might be a relation to your Thomas.
The other clue to follow up on is the address, in this case Carpenterstown townland, which is located about half way between the towns of Blanchardstown and Palmerstown. Carpenterstown is in the civil parish of Castleknock and Griffith’s Valuation took place here just before (late 1840s) we know your Cooke family were there, but shows no sign of anyone named Cooke in Carperterstown or within the civil parish of Castleknock at the time. The valuation did not include every household, and it’s possible they were there but not included due to living circumstances, or only passing through, we know they later moved on to Dublin city soon after the birth of John... A search for Fitzpatrick properties in the parish shows a Patrick at the town of Castleknock, which is a short distance to the east of Carpenterstown townland.
A search of the registers for the parish shows just one Cook baptism, back in 1799, and four marriages two before that for Thomas and Elizabeth, so it looking to me as though Thomas may have come from elsewhere.
Marriages generally took place in the bride’s current parish, and a search shows about 40 Fitzpatrick baptisms in Blanchardstown records, which are worth checking out to see if it’s possible to put together separate family groups, one of which might be Elizabeth’s family.
Carpenterstown townland (c1840 - OSI Historic Map) : http://maps.osi.ie/publicviewer/#V2,707641,737315,10,7
Did Jane travel alone to the US without parents or other family members, did she leave from Liverpool also ?
I can see the travel record for John age 17, occupation labourer arriving in Boston on his own. Based on details we have so far my suspicion is that one or both of the parents died.. another possible line of investigation may be finding the baptism or birth of Elizabeth jnr. No sign of her in Irish records so far, could she have been born in England, possibly Lancashire ?