I didn't read this as her volunteering him. She asked if he was going to the game (didn't assume it) and actually said that "one" of them would give the if a ride ( at least that's what RAI wrote.
The shared calendar is called Cozi, but it only works when both parties are amenable to using it.
RAI- the medical stuff should be specifically addressed as part of your settlement. In the meantime, I would try simply and politely stating that you would like to informed about all of the kid's doctors appt.
Frankly it sounds like you need a legal separation agreement sooner than later given what you've written about kids and finances.
