Statutory registration of births only started in Ireland in 1864 which is probably why you cannot find all the births on Family Search. You would need to search baptism records to find those pre 1864. These are not all on line but you can see what exists from the PRONI site. You need to knwo the family's exact religion:
For marriages, RC marriages were only registered from 1.1.1864 too. Protestant and nonconformist marriages were registered from 1.4.1845. I can’t see the Hazlett – Kane marriage in the civil registers, and suspect therefore it may have been RC. If so, again you would need to rely on parish records to trace it.