petting and bathing is coming for sure. I dunno about the idea of adopting or training them. Maybe. Not sure how that would work.
perhaps you can mark a specific girl as a favourite so when going to a location her species can go to it will always spawn her exact look/personality and she'll act familiar to you, as if you have gained her trust before. interact with the same girl enough times in different locations/times of day and maybe she'll follow you 'home'. you could get little accessories like ribbons or chokers and maybe send them out to explore for items or send her looking for specific monster gals to give a bonus boost to finding rarer ones.. and perhaps you can come across her in the location you sent her to for multigirl play... all just spitballin' ideas of course.