irishroots.ie is the website to check baptism records, but there are big gaps in the records for Co Sligo.
The Sligo Genealogy and Heritage Centre have a website and email address - they're very helpful.
James in GV is probably her father but I would eliminate other possibilities first, namely search baptism records on irishroots.ie all over Sligo for a Catherine with father James around 1852.
In the 1901 census there are a few Mulrenin families - type in Mulr* in the search box and it comes up with variants.
It might be worthwhile searching these families backwards - I'd start with those families with James as the eldest son...possibly named after his grandfather - the James in GV. It's a long shot but this is the type of thing that sometimes works.
Did she have any siblings you know of? Maybe possible to find some clues in their records.
Any Coleman connection can only be possible if they lived in neighbouring parishes on the Sligo/Roscommon border. This happened in my family with couples born in two different counties but actually within a mile of each other.