I think that you deserve a giant award for accepting the responsibility for your friend's child (and dog!) for a not short period of time.
I highly doubt that your friend will teach her boyfriend anything by showing up for his birthday other than that she will do things for him while he doesn't do for her. And as to asking you to watch her child and dog for the better part of a week? To me it seems like she is really taking advantage of you. I understand that she doesn't want to leave her child with someone she doesn't trust, and it's much easier to board a dog (in a kennel, that is) than find someone to care for a child properly - but it seems that she is totally escaping her own responsibilites. From your description it appears that she doesn't understand how to teach her own child (and dog) acceptable behaviors.
Unfortunately I don't have anything to offer in the area of how to nicely deal with this 'friend'. I have a feeling, based on her reaction to the dog's behavior, that she really doesn't have a clue as to how to deal with her own child or dog, and that she's not likely to accept constructive criticism. I agree with you that it is needed though.
I think your idea of not seeing her for a while (after she picks up child & dog, that is) is a good one.
--- Denise