Ellen Helen Brannigan, born 1817 Ireland, possibly County Monaghan

[Maiden Stone]

So are you happy with the finds in Drogheda, Co Louth made by Heywood and MS and the case is closed? 

There are a huge number of Brannigan links on this site that you could follow up to see if your Co Louth family is related further back, also some in Belfast
https://www.irishgenealogy.ie/en/
and of course through the Church registers that the wizards, Heywood and Maiden stone, have found.

We can leave it to you to follow up Thomas on the census and Catherine. 

There's no firm evidence to link the Brannigan family in the Drogheda church registers with the family of Peter and Ellen. It's just another family to test and probably eliminate. The Ellen who was baptised at Drogheda in 1822 was unlikely to have married in 1834 when she was 12. It's more probable that she was the same Ellen who married at Drogheda in 1846 aged 23 or 24. Likewise, the Catherine whose baptism heywood found may have been the Catherine who married at Drogheda in 1852.
Drogheda is one of a few parishes in County Louth which has registers dating from the start of the 19th century. We only considered Louth because it was Catherine Brannigan's birthplace. Catherine might be a red herring. There's no confirmation from lyroesesss that the Catherine Brannigan who died in NY belonged to Peter and Ellen's family. 
I looked up death registrations for Peter Brannigan and Mary Brannigan in Drogheda district and there were a few for Peter and many for Mary of the right age for parents of those baptisms. Civil registration of deaths began 1864.

County Monaghan, likewise, has only a few parishes with baptism registers dating back to the decades when Ellen and Peter were born.
Ellen and Peter were probably baptised in a parish for which baptism registers don't exist for those decades.

I don't do wizardry, shanreagh.  I only consider evidence, ask questions, and use logic, historical knowledge and experience.  Rootschatters are like a team of detectives, we share knowledge and test evidence.

[Maiden Stone]

 John A. Brannigan’s gravestone, although badly deteriorated, has his father and other family names attached.

https://www.findagrave.com/memorial/129269076/hn-a-branigan?_gl=1*xrcyj7*_ga*NzgyNDAyMTg5LjE1OTU3NDU3MDk.*_ga_4QT8FMEX30*MTY1MTM0NjI0Mi42LjEuMTY1MTM1MzU0My4w

An impressive memorial.
Do you know which other names are on it besides John and his wife?
Are there any other Brannigan graves nearby?

[shanreagh]

Yes MS and others my reply at was a little tongue in cheek.   

Unlike many families there seems to be lots of information on a number of Brannigan families both RC & CoI, across parts of Ireland that have relatively good records, not all the Counties do, and with a number of US records that are useful to check as they have good census records, not all countries do. 

OP the way to harness this information and that of the researchers on this thread who love researching, is to be as clear as you can about the records that you have and put them up here.  This means a stepped back process from the ancestors whose records you have no doubts about. That can be the starting point and then you can give us

the names you believe are their father mother siblings BDM, locations
aunts uncles
the names you believe are their grandfather, grandmother g/aunts & uncles

Or, seeing as we are looking at Ellen/Helen
1   What are her Appleton/Brannigan children's names in date order so BDM, locations
2    Do you know of any of the siblings of Ellen/Helen?  These would be the aunts and uncles of her Appleton/Brannigan children
3  What are the names of Ellen/Helen's grandchildren

Incidental aspects such as gravestone are very important.....sometimes they may give a birth  location.
So referring to John A Brannigan's gravestone you say it has his father and other family names attached but then do not list them. 

We want to help.  We like to work methodically as that will give us the best chance of a break through. Your knowledge of your family is essential and to have it written down means we can double check before going backwards or forwards and saves us going back over territory.  Our musings are an important part of that.

I am sure  we will get there.......!

ETA
Just looking at the Brannigan gravestone the link does give access to other brannigan burials here and several other NY cemeteries.  Have you looked at all of these to see if others of your known family are buried there?  There seem to be at least 58 others in this cemetery alone.