[LunarBerry]

I hear ya in the motherly instinct thing, I call my furbaby Kimchi my daughter and my mother refers herself as grandma, and my little niece thinks of her as a little cousin.
One thing that may help when leaving him alone is leaving him with something that has your scent on it, like a toy or maybe a piece of clothing that he can chew on safely. Also giving him and interactive toy, something that'll keep his attention.
As for the park off leash thing. I would never allow my girl off leash in open areas (only inclosed/fenced in) unless she was properly trained and even then, you'd have to consider that other people may not have dogs that are as nice or well behaved (also that people or wild animals may try and snatch her up). As pups they'll listen to you and not want to be far from your at first, especially in new places they are not familiar with, but they will get into the exploring mode the more comfortable and familiar the area is to them. So I wouldn't let her off leash anymore in any non-enclosed areas until she is at least properly trained to come at command, heel, etc. I'm fortunate that I have a doggie park at the place I take my girl to be groomed, they have a fenced-in large area I can let her off leash (which I'll only do if we're alone or with familiar dogs/people), otherwise, she stays on the leash (or as I prefer, harness).