Add my flabber to the already ghasted ones.
Holy crud monkeys, Batman!! What person in his or her right mind would think it was fine to take an injured child to Urgent Care and basically dump her there? Especially one that is as young as the OP's DD.
If this were my child, there would be no more play dates. I know the girls are friends, but so many red flags.
The other mother (now known as evil woman EW) takes offense that the OP has her cousin living with her and OP did not ask for her explicit permission.
EW will not allow the OP to speak to the situation, she shuts her down.
The apparent safety of EW's home is not so safe since she does not watch the girls.
EW thinks its okay to dump a small child with an injury at an UC and leave her there.
EW does not think its important to inform the OP of the injury.
I would be reporting the situation to CPS. This woman has major problems and I would not trust her with her own child.