It sounds to me like this most likely is this lady's dog and Nasdaq seems fairly confident in that as well. I trust her assessment of the situation. The suggestions being given--identifying marks, requiring proof via pictures, only giving her back if the dog acts excited, etc--seem a bit much, maybe?
Like false hope? (I dunno...hard to put into words.)
I know it must be a terribly hard thing to do to give up a dog when you've already gotten attached to it, spent money on vet bills, named it, bought it clothes, etc etc. But this was a lost dog from the very start only 2 weeks ago and it's important not to forget that.
I understand 'hoping' that the owner would let you keep her, even pay for her, but that's not likely to happen. How many members here would do that with one of theirs that had been lost just because the person who found them got attached?
Regardless, I think that "Little Mommy/Molly" was lucky to have been found by someone who gave her such loving care until her owner was found.