It's hard to tell how much of a problem there is and/or whether the woman is yanking your chain because she's jealous or unhappy with you or something else totally. It doesn't sound like she really wanted the dog in the first place or is interested in/knows how to take care of it well -- I would have immediately taken a dog someone gave me to my vet or its previous vet (if I didn't have a vet near me I knew about/wanted to use) to get it established and looked over and a baseline report, make sure its shots were up-to-date, transfer it's medical records, etc.

But if you have given her the dog and it doesn't sound like she's going to give it back, I'd limit the conversation so she doesn't tell you about the dog and upset you (since there really isn't anything you can do about it now?). If she asked me questions like with the fleas and doesn't understand my answers I'd suggest she ask the dog's vet and keep urging her to have the dog looked over by her vet (rather than talking to you). It sounds to me like she is trying to "hurt" you in some way by mentioning how the dog is doing poorly under her care when you can't do anything about it! I'd at least stop the passive aggressive talk if I couldn't get the dog back.