When you are not sure where in Ireland someone came from its difficult to make a start, so here is some general information.
There are Mulkerrin's in the telephone directory for Galway.
You could try writing a few of those and just two Conley's.
In 1856 Galway the Griffith Valuation took place.
There was Mulkerrin's in Galway at the time in the Parish of Moyrus, Union of Clifden. Did not find any Mulkerrin's in Lettermullen but the place name but a play close
There is a History Foundation with birth and marriage records, you might be lucky with born 1832 as records before that are sparse in all parish's. Barbara is not a common name for the times, so with a birth year 1832 and name Barbara Conly you should get a record, with her father
Hugh you can be sure to find one. Galway genealogically is
divided into east and west but on this database you can search all Ireland with just the date, name and father's name. A birth record will only give you her parents names which you have already but also place/parish and date. With both parents names you can do one search that should include all her siblings.
Michael is a pretty common name. I thought you could search for Coleman is it is fairly unusualy but not so much in
Galway. I found two Coleman's on www.familysearch.org
with father's Michael and mother's Barbara but wrong maiden names. However, you should easily get all children born in Ireland from the same Foundation source as you have both of their surnames.
Sometimes immigration dates in the census are off or different dates given on different census.
If any of the family were still in Galway in 1901/11 you can
see the census at www.census.nationalarchives.ie