Unfortunately, the Church of Ireland records for Crossmolina Union (where you found the info above) are almost all that we have pre-Famine to seek answers to your questions.
I very much doubt you will find anything older re parents etc. But use the following resources to see if you can get 'probable' matches for the family members.
You should also check out the surviving church of Ireland records for neighbouring areas of Killala, Ballysakeery & Kilmoremoy for the pre-Famine period, You may get lucky.
As for the origins https://www.johngrenham.com/findasurname.php?surname=Neely is a great resource.
I am almost certain that Neely is a Mac Neely originally who converted to Chuch of Ireland and dropped the Mac. They may have migrated with the other 'Ultaghs' or Ulster dispossed from the Donegal / Derry area after the Cromwellian wars of the 17th century. Unlike most of their comrades, they seem to have become protestants. This was not commonplace but did happen.
All the best