I sometimes lurk here but rarely butt in. In this case though...
Being a parent is bloody difficult sometimes, especially when children are ill. Different people, whether together or separated, have hugely different thresholds for when to seek medical advice. I tend to be much more cautious if I've not got another adult to bounce my concerns off.
Attacking this child's mother for her health-related decisions strikes me as cheap and unnecessary, and makes you look bad. She is communicating information and concerns re her child's health to her child's father - surely that is a good thing?
And to correct some misconceptions:
There is no cut-off age for the diagnosis of asthma.
If the child warranted a referral to hospital, and needed a chest x-ray and oxygen, she clearly was pretty poorly. And yes, she could have been sent home, especially if the hospital was under bed pressure. The doctors would have only allowed this if they thought the mum was sensible, had the logistics to be able to return if necessary, and the child appeared to be improving.
It is sometimes appropriate to take a child to a doctor for headache, It is definitely appropriate to ask about rashes (ever heard of meningitis?), head injuries, and any bugs going around.
From the health-related information you have posted, you start talking about Munchausens by proxy? Unbelievable. She sounds like a pretty on-the-ball sensible parent to me. Sorry.
I do appreciate you may have other reasons to dislike her, but you are going too far here.