By 1863, the power of the kirk session had waned considerably, especially in industrial towns that attracted "incomers", so I doubt you'd find that she'd been brought before the kirk session. In any case, the kirk session minutes can only be researched at selected archive centres in Scotland: they are not online as they haven't yet been indexed.
After birth certificates were introduced in 1855, registrars were only allowed to include the father's name for an illegitimate child if the father attended the registration or gave his written permission. They were not allowed to take the mother's word for it!
Sometimes an illegitimate child names their father on their marriage certificate but sadly your ancestor hasn't. And yes, a middle name can sometimes be the father's surname, but a girl named Elizabeth Stewart Moffat was more likely to have been named for someone called Elizabeth Stewart. If Elizabeth was her mother's first daughter then it could well be that her grandmother's maiden name was Elizabeth Stewart.
I have ancestors who lived at High Street, Kilmarnock. Most of the houses seem to have been multi-tenanted, with families living in one or two rooms. And yes, back in 1863 Scotland both births and marriages would have taken place at home, and not in a hospital or church.